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was utilized in the secondary-school textbook by
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to provide a new standard for teaching high school geometry, known as
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The non-Euclidean, hyperbolic plane: its structure and consistency
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on any line can be put into a 1:1 correspondence with the
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can be put into 1:1 correspondence with the real numbers
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Theoretical framework for planar
Euclidean geometry
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567:(3rd ed.), American Mathematical Society,
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587:Kelly, Paul Joseph; Matthews, Gordon (1981),
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203:that contains any two given distinct points
39:that can be confirmed experimentally with a
55:-based introduction to Euclidean geometry.
35:. These postulates are all based on basic
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297:of the numbers associated with the lines
219:Postulate III: Postulate of angle measure
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105:, and the angle formed by three points
31:in the plane, sometimes referred to as
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130:Postulate I: Postulate of line measure
47:. Since the postulates build upon the
359:Postulate IV: Postulate of similarity
197:. There is one and only one line
74:use variants of Birkhoff's axioms.
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195:Postulate II: Point-line postulate
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82:The distance between two points
70:. A few other textbooks in the
276:, respectively, the difference
51:, the approach is similar to a
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64:School Mathematics Study Group
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316:. Furthermore, if the point
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355:varies continuously also.
338:not containing the vertex
78:Birkhoff's Four Postulates
258:are points (not equal to
361:. Given two triangles
617:Foundations of geometry
483:Foundations of geometry
72:foundations of geometry
561:Birkhoff, George David
512:Birkhoff, George David
132:. The set of points
23:created a set of four
517:Annals of Mathematics
221:. The set of rays
622:Elementary geometry
591:, Springer-Verlag,
295: (mod 2π)
286: −
165: −
539:10338.dmlcz/147209
473:Euclidean geometry
373:and some constant
229:through any point
29:Euclidean geometry
574:978-0-8218-2101-5
433:B', C'
403:A', C'
378: > 0
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225:ℓ, m, n
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