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The Mathematics Teacher, 1922
About 2200 years ago there was published in Greek one of the most remarkable books of all times, Euclid's “Elements of Geometry”. It contains a systematic exposition of the leading propositions of elementary geometry and the elementary theory of numbers. It was at once adopted by the Greeks as the standard text book on pure mathematics.
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About 2200 years ago there was published in Greek one of the most remarkable books of all times, Euclid's “Elements of Geometry”. It contains a systematic exposition of the leading propositions of elementary geometry and the elementary theory of numbers. It was at once adopted by the Greeks as the standard text book on pure mathematics.
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1984
It is unlikely that Euclid ever held his five postulates to be self-evident. Mathematicians sharing the Aristotelian conviction that only manifest truths may be admitted without proof in geometry usually did not find the fifth postulate quite so obvious as the other four.
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It is unlikely that Euclid ever held his five postulates to be self-evident. Mathematicians sharing the Aristotelian conviction that only manifest truths may be admitted without proof in geometry usually did not find the fifth postulate quite so obvious as the other four.
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1997
Euclid’s Elements is dull, long-winded, and does not make explicit the fact that two circles can intersect, that a circle has an outside and an inside, that triangles can be turned over, and other assumptions essential to his system. By modern standards Bertrand Russell could call Euclid’s fourth proposition a ”tissue of nonsense” and declare it a ...
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Euclid’s Elements is dull, long-winded, and does not make explicit the fact that two circles can intersect, that a circle has an outside and an inside, that triangles can be turned over, and other assumptions essential to his system. By modern standards Bertrand Russell could call Euclid’s fourth proposition a ”tissue of nonsense” and declare it a ...
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1990
Abstract Amidst all the complex technical creations of the nineteenth century the most profound one, non-Euclidean geometry, was technically the simplest. This creation gave rise to important new branches of mathematics but its most significant implication is that it obliged mathematicians to revise radically their understanding of the ...
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Abstract Amidst all the complex technical creations of the nineteenth century the most profound one, non-Euclidean geometry, was technically the simplest. This creation gave rise to important new branches of mathematics but its most significant implication is that it obliged mathematicians to revise radically their understanding of the ...
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2020
One of the new frontiers in geometry opened up by calculus was the study of curvature. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding ...
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One of the new frontiers in geometry opened up by calculus was the study of curvature. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding ...
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1981
In Chapter 3 we saw that the group G which we have been discussing is formed from the transformation $$ y = \frac{{xT + xx'{v_1} + {v_2}}}{{xu{'_2} + xx'b + d}} $$ (1) (And at the same time we have $$ yy'\left( {\frac{{xu{'_1} + xx'a + c}}{{xu{'_2} + xx'b + d}}} \right).) $$ (2) Observe that the matrix $$ M = \left( {\begin ...
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In Chapter 3 we saw that the group G which we have been discussing is formed from the transformation $$ y = \frac{{xT + xx'{v_1} + {v_2}}}{{xu{'_2} + xx'b + d}} $$ (1) (And at the same time we have $$ yy'\left( {\frac{{xu{'_1} + xx'a + c}}{{xu{'_2} + xx'b + d}}} \right).) $$ (2) Observe that the matrix $$ M = \left( {\begin ...
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2011
In this chapter we discuss the non-Euclidean geometry of curved surfaces, using the sphere as our primary example. We find that all the information about the geometry of the surface is contained in the expression for the distance between two nearby points in some coordinate system, called the metric. For example, the distance between two distant points
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In this chapter we discuss the non-Euclidean geometry of curved surfaces, using the sphere as our primary example. We find that all the information about the geometry of the surface is contained in the expression for the distance between two nearby points in some coordinate system, called the metric. For example, the distance between two distant points
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2020
Abstract Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably ...
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Abstract Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably ...
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2014
The fruitless attempts to prove Euclid’s parallel postulate, in particular the theory of limit parallels, lead eventually the mathematicians of the nineteenth century to consider that the negation of this postulate could possibly be taken as an axiom.
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The fruitless attempts to prove Euclid’s parallel postulate, in particular the theory of limit parallels, lead eventually the mathematicians of the nineteenth century to consider that the negation of this postulate could possibly be taken as an axiom.
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