You must define what a line is carefully here. “Yeah, but these are not lines,” you may say. In fact, on the sphere, there are no parallel lines. You saw the two vertical lines (both perpendicular to the equator) are parallel at the equator but end up meeting each other at the North Pole. Now, using the example at the beginning of this section (the “triangle” on the sphere), you may be able to see that this postulate is not true on the sphere. The original statement is weird, right? This is exactly what caused the controversy and, eventually, a revolution. (Originally this statement was more like this: “If two lines both crossing another line form two interior angles on the same side whose sum is less than two right angles (180 degrees), then the two lines, when extended indefinitely on that side, will eventually meet.”) Given a line L and a point P that is not on the line, there is one and only one line through P parallel to L. (And they’d better be obvious to everyone so that no one would question them.) Here are the first four:Įuclid's Fifth Axiom (Parallel Postulate) Just as you cannot define every word you use (because each definition uses other words, each of which also needs to be defined using other words), you cannot prove everything some statements must be assumed true at the beginning. 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. \): On a sphere, the sum of the angles of a triangle is not equal to 180°.
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