aKimuya .M. Alex, bMunyambu .C. June

In the absence of a comprehensive geometrical perceptive on the nature of rational (commensurate) and
irrational (incommensurate) geometric magnitudes, a solution to the ages-old problem on the constructability
of magnitudes of the forms √2 and √2 3 (the square root of two and the cube root of two as used in modern
mathematics and sciences) would remain a mystery. The primary goal of this paper is to reveal a pure
geometrical proof for solving the construction of rational and irrational geometric magnitudes (those based on
straight lines) and refute the established notion that magnitudes of form √2 3 are not geometrically constructible.
The work also establishes a rigorous relationship between geometrical methods of proof as applied in Euclidean
geometry, and the non-Euclidean methods of proof, to correct a misconception governing the geometrical
understanding of irrational magnitudes (expressions). The established proof is based on a philosophical certainty
that "the algebraic notion of irrationality" is not a geometrical concept but rather, a misrepresentative language
used as a means of proof that a certain problem is geometrically impossible.


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