remark 7.82 What Archimedes proved

Rests on

• depends_on proposition 7.79 The polygon recursion ¶
  • depends_on definition 7.74 $\pi$ ¶
    • depends_on axiom 7.1 Completeness of $\R$ ¶
    • depends_on lemma 7.73 $\cos 2 < 0$ ¶
      • depends_on equation 7.53 eq:ana-cos-sin-series ¶
      • depends_on proposition 7.52 Alternating series test ¶
        • depends_on corollary A.46 Monotone convergence ¶
        • depends_on definition 7.45 Series ¶
        • proves proof ch:05-real-analysis@proof-33 ¶
      • proves proof ch:05-real-analysis@proof-48 ¶
    • depends_on theorem 7.23 Intermediate value theorem ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on equation 7.7 eq:ana-limit-left ¶
        • depends_on equation 7.5 eq:ana-limit-right ¶
      • proves proof ch:05-real-analysis@proof-8 ¶
    • depends_on theorem 7.51 Termwise differentiation ¶
      • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on definition 7.45 Series ¶ ↺
        • depends_on theorem 7.8 Cauchy criterion ¶
        • proves proof ch:05-real-analysis@proof-28 ¶
      • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on definition 7.20 Continuity at a point ¶ ↺
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-11 ¶
      • depends_on proposition 7.48 Ratio test ¶
        • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶
        • depends_on theorem 7.8 Cauchy criterion ¶ ↺
        • proves proof ch:05-real-analysis@proof-29 ¶
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • depends_on theorem 7.50 Power series; radius of convergence ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶ ↺
        • proves proof ch:05-real-analysis@proof-31 ¶
      • proves proof ch:05-real-analysis@proof-32 ¶
  • depends_on lemma 7.78 Chord, arc, tangent ¶
    • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-20 ¶
    • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
      • depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
      • depends_on equation 5.3 eq:lin-binomial ¶
      • depends_on proposition 7.49 Cauchy product ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on definition 7.45 Series ¶ ↺
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-30 ¶
      • depends_on theorem 7.51 Termwise differentiation ¶ ↺
      • proves proof ch:05-real-analysis@proof-46 ¶
    • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      • depends_on definition 7.74 $\pi$ ¶ ↺
      • depends_on equation 7.58 eq:ana-half-angle ¶
      • depends_on lemma 7.65 The integer part ¶
        • depends_on axiom 3.33 Peano axioms ¶
        • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on lemma 7.64 Well-ordering of $\N$ ¶
        • proves proof ch:05-real-analysis@proof-42 ¶
      • depends_on lemma 7.75 The first quadrant ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
        • depends_on definition 7.74 $\pi$ ¶ ↺
        • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
        • depends_on theorem 7.23 Intermediate value theorem ¶ ↺
        • depends_on theorem 7.51 Termwise differentiation ¶ ↺
        • proves proof ch:05-real-analysis@proof-49 ¶
      • depends_on lemma 7.72 Addition theorems ¶
        • depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
        • depends_on equation 5.3 eq:lin-binomial ¶ ↺
        • depends_on proposition 7.49 Cauchy product ¶ ↺
        • depends_on proposition 7.48 Ratio test ¶ ↺
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-47 ¶
      • proves proof ch:05-real-analysis@proof-51 ¶
    • proves proof ch:05-real-analysis@proof-52 ¶
  • depends_on lemma 7.72 Addition theorems ¶ ↺
  • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
  • proves proof ch:05-real-analysis@proof-53 ¶

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