![]() What is an axiom? It is a (self-evident) statement assumed true without proof. In fact, he started with only five axioms. That whole thing-which is the fundamental structure of mathematics-was first established by Euclid.Īnother amazing accomplishment of Euclid was that he proved tons of propositions-465 to be exact-based on a very small number of assumptions. If you remember your high school geometry, you may recall memorizing postulates (general assumptions) and proving theorems based on known properties and other theorems. Well, for one thing, it was the first book that laid the foundation of deductive logic-to prove general statements (called propositions) by definitions, general assumptions, and already known propositions. He could have been quite wealthy all the royalties he could have earned (except he would not have cared-there is a well-known story of Euclid embarrassing and humiliating one of his students who wanted to know what he would gain by learning geometry). It’s too bad that the notions of copyrights and intellectual properties did not exist back then. It was the standard book in geometry for over 2000 years, and there are over 1000 editions of the book in hundreds of languages. This book may be the most widely read treatise in world history because no other books have been read longer or by more people, with the exception of the Bible. The reader is encouraged to find out more by doing a search under “non-Euclidean geometry.”Įuclid, who lived around 300 B.C., is best known for his book The Elements, a 13- volume masterpiece laying the foundations of geometry (and some number theory as well). Here, a very abbreviated version of the story is presented. What many experts feel is offensive and repugnant may actually be true.What stands the “test of time” may not be absolutely true.Common sense could be the greatest obstacle to finding truth.The story is worthy of a movie or a play. The conception and arrival of non-Euclidean geometry involved three mathematicians-one very famous and two completely unknown. ![]() This subject, “Euclidean geometry” (the type of geometry you studied in high school), was so popular and dominant that no one, for over two millennia, doubted its truthfulness, questioned its authority, or thought of coming up with an alternative. ![]() You see, Euclid (who lived over 2300 years ago!) wrote a textbook that was so popular that practically every educated person in the world used it to study geometry for the next 2000 plus years. It was truly a ground-shaking event, not only in the history of mathematics and but also in philosophy. The birth of non-Euclidean geometry was REALLY a big deal. It is called "Non-Euclidean" because it is different from Euclidean geometry, which was developed by an ancient Greek mathematician called Euclid. Image is used under a CC BY-SA 3.0 license. ![]() In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. The surface of a sphere is not a Euclidean plane, but locally the laws of the Euclidean geometry are good approximations. What the Geometry of Meaning provides is a much-needed exploration of computational techniques to represent meaning and of the conceptual spaces on which these representations are founded.\): On a sphere, the sum of the angles of a triangle is not equal to 180°. The rich geometry of conceptual space can be glimpsed, for instance, in internet documents: while the documents themselves define a structure of visual layouts and point-to-point links, search engines create an additional structure by matching keywords to nearby documents in a spatial arrangement of content. Currently, similar geometric models are being applied to another type of space-the conceptual space of information and meaning, where the contributions of Pythagoras and Einstein are a part of the landscape itself. From Pythagoras’s harmonic sequence to Einstein’s theory of relativity, geometric models of position, proximity, ratio, and the underlying properties of physical space have provided us with powerful ideas and accurate scientific tools. ![]()
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