All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident: Arthur Schopenhauer -- In questions of science the authority of a thousand is not worth the humble reasoning of a single individual: Galileo Galilei -- Science is a wonderful thing if one does not have to earn one's living at it: Albert Einstein -- When you have eliminated the impossible, what ever remains, however improbable must be the truth: Sir Arthur Conan Doyle -- We all agree that your theory is crazy, but is it crazy enough? Niels Bohr -- Whenever a true theory appears, it will be its own evidence. Its test is that it will explain all phenomena: Ralph Waldo Emerson -- Since the mathematicians invaded Relativity, I do not understand it myself anymore: Albert Einstein -- I would say that the aether is a medium invented by man for the purpose of propagating his misconceptions from one place to another: W.F.G. Swann: -- Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone: Albert Einstein -- Physics is mathematical not because we know so much about the physical world, but because we know so little: Bertrand Russell -- If I could explain it to the average person, I would not have been worth the Nobel Prize: R. P. Feynman -- I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use: Galileo Galilei -- How dare we speak of the laws of chance? Is not chance the antithesis of all law?: Bertrand Russell -- Only two things are infinite, the universe and human stupidity, and I´m not sure about the former: Albert Einstein -- The glory of mathematics is that you don't have to say what you are talking about: Richard Feynman -- Anything is possible if you don´t know what you are talking about: Author Unknown -- In life, everything is relative - except Einstein´s theory: Leonid S. Sukhorukov -- Don´\'t worry about people stealing your ideas. If your ideas are any good, you´ll have to ram them down people´s throats: Howard Aiken --A day will come undoubtedly when the ether will be discarded as useless: H. Poincaré -- First they tell you you´re wrong and they can prove it; then they tell you you´re right but it isn´t important; then they tell you it´s important but they knew it all along: Charles Kettering -- It is not once nor twice but times without number that the same ideas make their appearance in the world: Aristotle -- The opposite of a true statement is a false statement. The opposite of a profound truth may well be another profound truth: Niels Bohr -- A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it: Max Planck -- Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions: Eric Temple Bell -- Half this game is ninety percent mental: Yogi Berra

Proving Unproved Euclidean Propositions on a New Foundational Basis


Leon, Antonio


Research Papers


Mathematics and Applied Mathematics



Date Published:

September 21, 2020




foundation of Euclidean geometry, sidedness, straightness, orthogonality, parallelism, convergence.


This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.


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