A Rigorous Geometrical Proof on Constructability of Magnitudes (A Classical Geometric Solution for the Factors: √2 and √(3&2))


  • Munyambu .C. June Meru University of Science and Technology, Kenya
  • Kimuya Alex Meru University of Science and Technology


Delian Constant, Pythagorean Factor, Constructible Magnitudes, Constructible Points, Rational Numbers, Irrational Numbers, Euclidean Geometry, Doubling A Square, Constructability


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  and  (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  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.


Download data is not yet available.

Author Biographies

Munyambu .C. June, Meru University of Science and Technology, Kenya

Department of Education Science (Physics/Mathematics)

Kimuya Alex, Meru University of Science and Technology

Department of Physical Sciences (Physics)


T. L. Heath, The Thirteen Books of the Elements, Vol. 1: Books 1-2 (first published 1925), 2nd ed. New York: Dover Publications, 1956.

Heath, Thomas L, The Thirteen Books of Euclid’s Elements, translated from the text of Heiberg, with introduction and commentary, 2nd ed., vol. 3 vols. Cambridge (available in Dover reprint): University Press, 1926.

H. E. DUDENEY, The Canterbury Puzzles, and Other Curious Problems, 3rd ed. London: Thomas Nelson and Sons, 1929.

H. Dorrie, 100 Great Problems of Elementary Mathematics: Their History and Solution. New York: Dover Publications Inc., 1965.

H. BILLINGSLEY, The Elements of Geometrie, of the Most Ancient Philosopher Euclide of Megara. London: John Daye, 1570.

RRICHARD DEDEKIND, Stetigkeit und Irrationale Zahlen. Braunschweig, 1872.

DAVID HILBERT, Foundations of Geometry (first published as Grundlagen der Geometrie, 1899), 2nd ed. La Salle: Open Court, 1971.

K. M. A. Mutembei Josephine, “The Cube Duplication Solution (A Compassstraightedge(Ruler) Construction),” Int. J. Math. Trends Technol. IJMTT, Accessed: Apr. 07, 2020. [Online]. Available: http://www.ijmttjournal.org/archive/ijmtt-v50p549.

Pierre Laurent Wantzel, “Recherches sur les moyens de reconnaitre si un problème de géométrie peut se résoudre avec la règle et le compass,” Journal de Mathematiques pures et appliques, vol. 2, pp. 366–372, 1837, Accessed: Apr. 07, 2020. [Online]. Available: http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16381&Deb=374&Fin=380&E=PDF.

Herstein. I. N, Abstract Algebra. Macmillan, 1986.

Herstein, I.N. & Milnor, J.W., An Axiomatic Approach to Measurable Utility. 1953.

Aristotle, Metaphysic (428/27-348/47 BCE), Translated by W. D. Ross. 350AD.

Apostol, Tom M., One-Variable Calculus, with an Introduction to Linear Algebra, vol. 1. Indiana University: George Springer, 1976.


N. D. Kazarinoff, Ruler and the Round: Classic Problems in Geometric Constructions. Mineola, N.Y: Dover Publications, 2011.

Kazarinoff, Nicholas D., Ruler and the Round /Classic Problems in Geometric Constructions (first published 1970). Dover, 2003.

Robin Hartshorne, “Teaching Geometry According to Euclid,” Berkeley Not. AMS, 2000.

“Oxford advanced learner’s dictionary, seventh edition.” Oxford university press, 2007.

Johnstone. B, Discourse analysis. Blackwell Publishing Ltd, 2008.

Rene’ Descartes, Geometria, à Renato des Cartes anno 1637 Gallice edita: postea autem una cum notis Florimondi de Beaune,...in linguam latinam versa et commentariis illustrata opera atque studio Francisci à Schooten. Amstelaedami: apud Ludovicum et Danielem Elzevirios, 1659.

Rene’ Descartes, Renati Descartes Geometria. F. Knoch. 1695.

Clagett Marshall, Archimedes in the Middle Ages. Five Volumes. American Philosophical Society, 1964.




How to Cite

Munyambu .C. June, & Kimuya Alex. (2020). A Rigorous Geometrical Proof on Constructability of Magnitudes (A Classical Geometric Solution for the Factors: √2 and √(3&2)). MathLAB Journal, 5, 117-142. Retrieved from https://purkh.com/index.php/mathlab/article/view/730



Research Articles