By the seventeenth century additional speculative layers of a more esoteric nature had been realized in the theorem. Here we find the second layer of meaning added to this geometrical symbol.

Let us replace the base and the number 3 with the term Entered Apprentice; replace the perpendicular and the number 4 with the term Fellow Craft, and replace the hypotenuse and the number 5 with term Master Mason. Can we create a definition whereby these terms have substance in the context of the geometrical design? How might the Entered Apprentice correlate with the base, the Fellow Craft with the perpendicular, and the Master Mason with the hypotenuse?

Clearly the base is the shortest of the three lines and accordingly is assigned the shortest of the three dimensions. In this respect it correlates with the lowest degree in Symbolic Masonry. The perpendicular is in length assigned the number 4 and represents a measurement that is between the Entered Apprentice (the base) and the Master Mason (the hypotenuse) degree and therefore the perpendicular clearly represents the Fellow Craft in this paradigm. The hypotenuse is presented in the theorem as the number 5 representing the longest or greatest measurement of the right angle triangle. The theorem states that the determination of the hypotenuse is dependent on the base and the perpendicular. Within the ritual the third degree is clearly dependent upon the candidate being initiated and passed before he can attain the most valuable third degree. Thus, we can with confidence transpose each of the three degrees for the assigned numerical measurements given in the theorem, and thereby represent the three degrees of Freemasonry.

The transposition model enhances the 47th Problem of Euclid considerably from the perspective of a ritualistic assignment where we find it described within the work and apparent in how the Lesser Lights are laid out upon the floor of the Lodge. It still however, leaves us with a cloudy understanding of the meaning imparted in this important symbol. Therefore, the terms Entered Apprentice, Fellow Craft, and Master Mason must still be expanded further in the crafting of the Euclidic Paradigm.

In considering the term Entered Apprentice we come to appreciate the Entered Apprentice as the lower form of the Masonic mystery. In the operative period the Entered Apprentice was the physical laborer, the bearer of burden, the mason that worked with his back rather than his brain. He was given orders and expected to submit to the lower levels of labor and required direct supervision due to his lack of knowledge. His degree of understanding for the theory of masonry was little to none. Considering these characteristics, we might easily today represent the Entered Apprentice as the physicality, the body, or the brawn, one without a depth of knowledge, at least as far as the mysteries of the Craft are concerned. We can associate the Entered Apprentice in allegorical form as that which is focused on the vanity of physical appearance and demands, the needs and drives of the body, the contentiousness of mankind, and the self-focus. This is not to be thought of as a negative in anyway since it is the physical body as the ignorant laborer that houses that higher component, which we call the intellect. It is a third of the components that comprise the divine creation of humanity and a needed component in an understanding of who we really are. Given our physical natures we are all sometimes still, Entered Apprentices.

Examination of the Fellow Craft indicates that these were Masons of the operative period that could think and plan, make decisions without over the shoulder supervision, that were not yet masterful, but could nonetheless combine the talents of mind with those of body to accomplish a masonry construction. Within the speculative form, the Fellow Craft degree represents an unequivocal intellectual capability as is defined in the components of the Stairway Lecture. The Fellow Craft has learned to subordinate his physical demands and direct himself in an intelligent manner in order to accomplish some particular end. Within the speculative framework, the Fellow Craft unequivocally represents the uprightness of the perpendicular.

The geometrical theorem tells us that the hypotenuse is the combination of the sum of the square of the base plus the sum of the square of the perpendicular. If the hypotenuse does in truth represent the third degree or the Master Mason then we can inferentially reason that a Master Mason is one who has learned to perfectly couple his physicality with his intellectual skills and can arrive at the theoretical square of his whole self, but moreover can distill that down to his true identity, which is his own divinity represented by the inducted square root. The true Master Mason moves masterfully within his temporal existence, which is to say that he is in perfect balance in his own duality.

It is apparent from the transposition model that a substantive transition can be made from the geometrical theorem to a Masonic degree diagram. In expanding order then, we have a right angle triangle represented by the numerical values of 3, 4, and 5, followed by the terms base, perpendicular, and hypotenuse, which can be transliterated into Entered Apprentice, Fellow Craft, and Master Mason. By characterizing the degree names with terms inherent in each of the three degrees we begin to see the components of the Masonic characterization of our human sojourn, those being the physical, the intellectual, and the spiritual element or divinity of man.

The Euclidic Paradigm of Freemasonry then, qualifies for each Mason the components of temporal life. It would seem that sometimes we are in the mode of the Entered Apprentice when we are “me focused.” When the various passions of life dictate one’s behavior or interpersonal transactions, we are likely being ruled as the Entered Apprentice. Most conflicts are the result of two or more people focused on what they individually want. Their view is circumscribed solely to themselves in blindness to their brotherhood as originating from the One. A basis in fact can be presumed here as the Entered Apprentice degree teaches first and foremost the need to subdue one’s passions. This does not solely imply the passions as so often interpreted to mean sexuality, but in a much broader sense the passions of all physicality, the assumptions of the validity of material existence. When the self is demanding and unyielding, it is almost a certainty that the Entered Apprentice is ruling, lacking in awareness and understanding that this world is largely transitory and some would say fully illusionary, and insisting on a materialistic one-dimensional view with the self view as the foremost objective. Balance between physicality and reason is woefully lacking.

Within the paradigm it is clear with even a casual notice that he who considers carefully, expresses thoughtfully, and calms conflict is one who has learned to balance the physical nature with intelligent reasoning. He is representative of the sum of the square of his laborer self plus the sum of the square of his intellectual or thoughtful self, and is more of a two dimensional expression within the temporal experience.

Occasionally, we can succumb to pure intellectualization just as we repeatedly fall into the trap of the “me-focus.” Therefore, even in the mode of Fellow Craft, we can become one dimensional as the expression indicates, “all talk, no action.” The true Fellow Craft has a comfortable balance between thought (intelligence) and action (labor) and is therefore bi-dimensional.

The ritual informs us that it is unknown as to whether or not any of us will ever truly be a Master Mason. How can this be? In understanding the Euclidic Paradigm we of necessity must understand the geometrical theorem first. From the calculation of the sum of the square of the base plus the sum of the square of the perpendicular, we only arrive at the square of the hypotenuse, not the hypotenuse per se, that requires yet another step, i.e. solving for the square root. There is nothing in the theorem that advises us to calculate the square root of that square, which in esoteric terms, must be self-evident. He who explores must choose to take the final step. The ritual work gives us all three dimensions as 3, 4, and 5, but in fact the theorem requires that we actually find the square root in order to calculate the hypotenuse. Within the Euclidic Paradigm it is a clear suggestion that each Master Mason must find it for himself as the total sum of all his parts.

By examining the idea of “hope of redemption” we must necessarily conclude an intellectual process, since there cannot be a concept of hope without some contemplation of a need for something better. By implication we surmise that one has then determined through thought, that something has been analyzed or considered and therefore, hopes for something more. Therefore in the Christian model we find the second component of the paradigm consisting of the intellectual process deemed within the ritual work of Masonry as the Fellow Craft and corresponding to the geometrical perpendicular.

Islam assists us in gaining insight into the oneness of God, and here we find an example where the concepts of the base and the perpendicular become united as represented by the sum of the squares of the base and the perpendicular. Stated according to the Euclidic paradigm, it is a unification and perfect balance of the physicality, intellectual reasoning, and the spirituality of man all clearly recognizable as the greater and lesser mysteries or capacities contained within man and yet established in perfect balance.

The paradigm is further supported by the additional hidden mysteries of Masonry, which inform us of our true origin and destination, the reasons for our presence here, and the explanations for the Masonic tools and skills, which we must learn to use and understand in order to perfect the unity and balance of a coherent disequilibria in a tri-fold humanity.

The Euclidic Paradigm then consists of five overlapping right angle triangles. The first provides the numerical sequence 3, 4, and 5. The second defines the terms base, perpendicular, and hypotenuse. The third transposes those terms to Enter Apprentice, Fellow Craft, and Master Mason. In the fourth, the degree names are defined by their nature. In the fifth and last, three of the world’s major religions, all originating with Abraham and defining the balance and duality necessary to attaining the degree of Master Mason are identified. By definition of language, a Master Mason is one who is masterful, meaning masterful of the self as the great end of our Order. Mastery may only be achieved through a realization of our tri-fold humanity, body, mind, and spirit with spirituality being that part for which we solve in identifying the hypotenuse. Given the number of sides in the right angle triangle and the number of triangular layers within the Euclidic Paradigm, it becomes apparent that the paradigm also represents for us the symbolic name of Deity, which in this interpretation is taken to represent the divine spark within each Mason and therefore, qualifies the paradigm even further.

Through an understanding of the underlying symbolism contained within a simple geometrical form, the Master Mason can come to a realization of his identity in appreciating who he really is while ever continuing to use all the tools given him in order to integrate his thoughts and actions for the sacred purpose to which he has been initiated, passed, and raised. That purpose is without doubt stated and enacted within our ritualistic portrayals and symbols of ancient truths which tell us that we are spiritual beings first and come here of our own free will and accord to create. We are given the tools to navigate the temporal world while simultaneously told the means to remain connected to our source. All then concludes with that heinous culmination feared by the unenlightened and which is the grand fiction made horrid by the ignorance of the multitude when in truth, it is a mere return to place from which we came.