Ad Quadratum Construction and Study of the Regular Polyhedra

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fundamental elements of the four mathematical sciences.
A simple ad triangulum diagram can also be derived to give us the essential structure of
the trivium as we shall presently see.
These adquadratum and ad triangulum diagrams are doubly important and symbolic to us
for medieval architects and master-masons relied on similar constructions for their
designs.
The Millennium sphere crystallizes these constructs in its structure.
The ad quadratum and ad triangulum construction are therefore literally ex-planations,
i.e., renderings into planes of multidimensional events or structures. The Trivium (which
received its name like the quadrivium only much later with Boethius) has roots going
back to at least the fifth century B.C. 63 With the growth of democracy, oratory in fifth
century Greece had assumed an increased preponderance. Sophists, the speech writers,
coaches, and handlers of the day, more intent like those of today on persuasion than on
truth, opened schools of rhetoric with claims to teach oratorical skills, with Isocrates, the
first on record, in the fourth Century B.C. It is in fact in conscious opposition to their
influence and to counter their moral relativism that Plato founded his academy.
“For Plato, words express the essence of things grasped in
thought as concepts. Words are combined into sentences
(or prepositions) reflecting the necessary connection in
reality. This, in opposition to the sophists’ view that
language was merely conventional.” 64
However, it is with the Stoics (3 rd C. B.C.) that the trivium as such (though not in name
yet) emerges. For them logic which they divided into dialectic, grammar, and rhetoric
formed a unity and a branch of philosophy along with physics and ethics. Rhetoric
sought to discover linguistic means of persuasion for all arguments, strategies that would
work in any disputation while grammar explained the structure of language. Crates, a
stoic of the first century B.C. is credited with writing the first systematic Greek grammar.
We may note here that in India, Pa nini had conceived with his Sanskrit generative and
prescriptive grammar a much more elaborate system at least three hundred years before.
From the Greeks to the Latins through Martianus Capella and Pliny the Elder (first
Century A.D.) to the encyclopedists of the fourth and fifth centuries (Donatus,
Chalcidius, Boethius) and later, Isidore of Seville (d. 636), one reaches the ninth Century
Carolingian Renaissance with Alcuin and Scotus, and from them, through the Cathedral
Schools such as Chartres, the 12 th century renaissance and the foundation of the
universities. Here the liberal arts are fully developed, mastered and clearly become
preparatory to the study of philosophy and theology. This foundation of the liberal arts
63 David Wagner: op. cit.
64 ibid.