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12-hour clock abstract angle answer axiom systems Babylonian-Egyptian cat commune Chapter circle compute COUNTERESSAY create decimal determine distance earth elementary equal Euclid Euclidean geometry exactly example experience Explain fact feet Figure 11 Figure 17 Figure 9 finite formula fractions Geometry Figure Greek hypotenuse idea inches infinite collections infinite number infinity knowledge length linear logically necessary look mathematicians mathematics measure miles moon multiply myth-model nature negative numbers notion number system objects odd number parallel lines parallel postulate pattern physical world possible prime numbers problem procedure proof prove Pythagorean Pythagorean Theorem quadratic equation rational numbers real numbers real world reason relation Riemannian geometry right triangle rule Section VI Geometry Section VII Non-Euclidean self-evident side simple solution solve subtract Suppose symbols Table theorems TIDBIT tion trigonometry true truths VII Non-Euclidean Geometry whole numbers zero