Acknowledgements |
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xxi | |
Prologue |
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xxiii | |
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The beginnings and the writing of the Principia |
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1 | (16) |
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1 | (1) |
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1 | (6) |
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7 | (1) |
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7 | (3) |
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The years 1685--1686: the writing of the Principia |
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10 | (7) |
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Basic concepts: Definitions and Axioms |
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17 | (26) |
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17 | (1) |
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Basic concepts: Definitions |
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17 | (5) |
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17 | (1) |
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18 | (1) |
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18 | (1) |
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19 | (1) |
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19 | (1) |
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20 | (1) |
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20 | (1) |
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20 | (2) |
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Basic concepts: the Laws of Motion |
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22 | (8) |
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22 | (1) |
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23 | (1) |
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23 | (1) |
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24 | (3) |
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27 | (2) |
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29 | (1) |
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The Scholium to the Laws of Motion |
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30 | (5) |
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Additional amplifications |
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35 | (8) |
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The proportionality of mass and weight and the experiments on the pendulums |
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35 | (1) |
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Proposition XXIV, Book II |
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35 | (2) |
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Maxwell's reformulation of Newton's Laws of Motion |
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37 | (4) |
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The Newtonian principle of relativity |
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41 | (2) |
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On the notion of limits and the ratios of evanescent quantities |
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43 | (14) |
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43 | (1) |
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43 | (1) |
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44 | (3) |
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47 | (3) |
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50 | (2) |
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52 | (5) |
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On the motion of particles under centripetal attraction: an introduction to Newton's treatment |
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57 | (10) |
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57 | (1) |
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The dynamics of a particle under a general law of centripetal attraction |
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58 | (3) |
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The conservation of angular momentum |
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58 | (1) |
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59 | (1) |
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The conservation of energy |
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60 | (1) |
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The equation governing r in the orbital plane |
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60 | (1) |
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The dynamics of a particle under the inverse-square law of attraction |
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61 | (3) |
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The Lenz vector and the Lenz equation |
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62 | (1) |
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63 | (1) |
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An alternative derivation of the elliptical orbit |
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64 | (1) |
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The accelerations and velocities along a curved orbit |
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64 | (3) |
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The law of areas and some relations which follow |
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67 | (26) |
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67 | (1) |
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67 | (9) |
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67 | (2) |
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69 | (1) |
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70 | (1) |
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70 | (1) |
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71 | (1) |
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71 | (1) |
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72 | (1) |
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73 | (2) |
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75 | (1) |
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Newton's relations for determining the law of centripetal attraction from the orbit |
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76 | (7) |
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76 | (1) |
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77 | (2) |
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79 | (1) |
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80 | (1) |
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80 | (2) |
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82 | (1) |
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Two simple illustrations of the basic relation |
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83 | (10) |
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83 | (2) |
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85 | (1) |
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86 | (1) |
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86 | (2) |
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88 | (1) |
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89 | (4) |
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The motion of bodies along conic sections |
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93 | (21) |
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93 | (1) |
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93 | (4) |
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96 | (1) |
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97 | (1) |
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98 | (1) |
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Proposition XIII: the motion of a body along a parabola |
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98 | (5) |
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98 | (2) |
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100 | (2) |
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102 | (1) |
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Kepler's third law: Propositions XIV and XV |
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103 | (1) |
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103 | (1) |
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104 | (1) |
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104 | (1) |
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Amplifications: Proposition XVI |
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104 | (10) |
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104 | (1) |
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105 | (2) |
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107 | (3) |
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110 | (2) |
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112 | (1) |
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113 | (1) |
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Supplement: on dual laws of centripetal attraction |
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114 | (229) |
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114 | (5) |
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115 | (1) |
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The orbit described is an ellipse |
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115 | (1) |
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The orbit described is a hyperbola |
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116 | (1) |
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A body orbiting the conjugate hyperbola with the centre of attraction at S |
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117 | (1) |
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The self-duality of the inverse-fifth power law of attraction |
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117 | (2) |
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The mapping of orbits described in the complex plane |
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119 | (3) |
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The dual laws of centripetal forces |
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122 | (5) |
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Kepler's equation and its solution |
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127 | (16) |
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127 | (1) |
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128 | (2) |
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130 | (3) |
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132 | (1) |
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133 | (6) |
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139 | (4) |
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141 | (2) |
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The rectilinear ascent and descent of bodies |
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143 | (22) |
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143 | (1) |
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An ab initio treatment of rectilinear motion |
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143 | (5) |
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144 | (1) |
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145 | (2) |
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147 | (1) |
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148 | (1) |
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148 | (2) |
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150 | (4) |
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153 | (1) |
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153 | (1) |
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The reduction of the problem of rectilinear motion to one in circular motion |
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154 | (6) |
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154 | (1) |
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155 | (1) |
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156 | (1) |
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157 | (1) |
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158 | (2) |
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160 | (1) |
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160 | (1) |
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161 | (1) |
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The initial-value problem |
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161 | (4) |
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161 | (2) |
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163 | (2) |
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The conservation of energy and the initial-value problem |
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165 | (18) |
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165 | (1) |
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165 | (3) |
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165 | (3) |
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168 | (1) |
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168 | (4) |
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172 | (1) |
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The mystery of the missing corollary |
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172 | (2) |
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Motion under an inverse-cube law of centripetal attraction |
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174 | (6) |
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179 | (1) |
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180 | (3) |
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183 | (18) |
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183 | (1) |
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The theorem of revolving orbits |
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184 | (1) |
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184 | (3) |
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187 | (5) |
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189 | (3) |
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192 | (9) |
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194 | (2) |
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196 | (5) |
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201 | (4) |
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201 | (1) |
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201 | (4) |
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203 | (2) |
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205 | (14) |
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205 | (1) |
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The two-body problem: the general theorems |
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206 | (7) |
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206 | (2) |
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208 | (1) |
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209 | (1) |
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210 | (1) |
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211 | (1) |
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212 | (1) |
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213 | (2) |
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213 | (1) |
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214 | (1) |
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The solution of a many-body problem |
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215 | (4) |
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215 | (4) |
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The method of the variation of the elements of a Kepler orbit and Newton's lunar theory: an introduction to Propositions LXV--LXIX |
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219 | (16) |
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219 | (1) |
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The basic equations, definitions, and the coordinate system adopted |
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219 | (4) |
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The variation of the elements |
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223 | (5) |
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223 | (1) |
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223 | (1) |
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224 | (1) |
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224 | (1) |
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225 | (1) |
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226 | (1) |
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Variation of Kepler's equation |
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227 | (1) |
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228 | (1) |
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Application of the method of the variation of the elements to lunar motion |
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228 | (7) |
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229 | (1) |
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The components of F( = (Fr, Fα, Fh)) |
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230 | (3) |
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Application of the variational equations |
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233 | (2) |
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The three-body problem: the foundations of Newton's lunar theory |
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235 | (34) |
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235 | (1) |
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235 | (2) |
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236 | (1) |
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236 | (1) |
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237 | (2) |
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237 | (2) |
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Proposition LXVI (continued): Corollaries I--VI |
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239 | (8) |
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239 | (2) |
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The centripetal attraction |
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241 | (1) |
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242 | (1) |
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The variation of the `constant of areas' |
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243 | (1) |
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243 | (1) |
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244 | (3) |
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Proposition LXVI (continued): Corollaries VII and VIII---the rotation of the line of apsides |
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247 | (3) |
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247 | (3) |
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250 | (1) |
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Proposition LXVI (continued): Corollaries IX--XVII |
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250 | (9) |
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Corollary IX: the variation of the eccentricity |
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250 | (2) |
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Corollary X: the variation of the inclination |
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252 | (2) |
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Corollary XI: the variation of the direction of the ascending node (Ω) |
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254 | (1) |
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255 | (1) |
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Further elaborations: Corollaries XIII--XVII |
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256 | (1) |
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256 | (1) |
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257 | (1) |
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257 | (1) |
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257 | (1) |
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257 | (1) |
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258 | (1) |
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Proposition LXVI (continued): Corollaries XVIII--XXII |
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259 | (6) |
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260 | (1) |
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260 | (2) |
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262 | (1) |
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263 | (1) |
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263 | (2) |
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265 | (4) |
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265 | (1) |
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Proposition LXIX and Corollaries I and II |
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265 | (2) |
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267 | (1) |
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267 | (2) |
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269 | (34) |
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269 | (1) |
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269 | (1) |
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269 | (6) |
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269 | (1) |
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270 | (3) |
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273 | (1) |
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273 | (2) |
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275 | (1) |
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Propositions LXXIII--LXXV |
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275 | (5) |
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275 | (1) |
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276 | (2) |
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278 | (1) |
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278 | (1) |
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279 | (1) |
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280 | (3) |
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281 | (2) |
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Propositions LXXVII and LXXVIII |
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283 | (4) |
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283 | (1) |
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283 | (3) |
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286 | (1) |
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286 | (1) |
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286 | (1) |
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Lemma XXIX and Propositions LXXIX--LXXXI |
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287 | (6) |
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291 | (2) |
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Proposition LXXXII and the discovery of the method of inversion and the principle of images |
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293 | (5) |
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294 | (2) |
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On the discovery of the method of images |
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296 | (1) |
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297 | (1) |
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Propositions LXXXIII and LXXXIV |
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298 | (4) |
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298 | (1) |
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299 | (1) |
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299 | (3) |
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Some personal reflections |
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302 | (1) |
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Attraction by non-spherical bodies |
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303 | (20) |
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303 | (1) |
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How we may discriminate between different laws of centripetal attraction |
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303 | (3) |
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304 | (1) |
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305 | (1) |
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306 | (1) |
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Proposition LXXXVII and Corollary I |
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306 | (1) |
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307 | (1) |
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Propositions LXXXVIII and LXXXIX |
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307 | (2) |
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307 | (1) |
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308 | (1) |
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308 | (1) |
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309 | (1) |
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The attraction by circular discs and round solids at points along their axes |
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309 | (4) |
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309 | (1) |
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310 | (1) |
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311 | (1) |
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312 | (1) |
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Corollaries II and III of Proposition XCI and Proposition XCII |
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313 | (4) |
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313 | (3) |
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316 | (1) |
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317 | (1) |
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317 | (6) |
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318 | (1) |
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319 | (1) |
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320 | (3) |
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A digression into Opticks |
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323 | (20) |
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323 | (1) |
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323 | (4) |
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324 | (2) |
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326 | (1) |
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326 | (1) |
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327 | (2) |
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329 | (5) |
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329 | (1) |
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330 | (1) |
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331 | (1) |
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331 | (1) |
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332 | (2) |
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The concluding Scholium of Book I |
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334 | (9) |
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334 | (1) |
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Appendix I. An analytic solution for the ovals of Descartes |
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334 | (4) |
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Appendix II. Maxwell on the ovals of Descartes |
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338 | (2) |
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340 | (3) |
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Introduction to Newton's System of the World (Book III) |
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343 | (194) |
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345 | (8) |
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Rules of reasoning in philosophy |
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345 | (1) |
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346 | (2) |
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348 | (5) |
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The universal law of gravitation |
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353 | (28) |
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353 | (2) |
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Propositions and rules (to which references are made) |
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353 | (2) |
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355 | (2) |
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Proposition IV and the Moon test |
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357 | (4) |
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360 | (1) |
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The emergence of the law of gravitation |
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361 | (1) |
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361 | (1) |
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362 | (1) |
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Proposition VI: the confirmation of the equality of the inertial and the gravitational masses by astronomical data |
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362 | (8) |
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362 | (7) |
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369 | (1) |
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Proposition VII: the universal law of gravitation |
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370 | (1) |
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370 | (1) |
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Propositions VIII and IX: the implications of the `superb theorems' |
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371 | (4) |
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371 | (1) |
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372 | (2) |
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374 | (1) |
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374 | (1) |
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374 | (1) |
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374 | (1) |
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374 | (1) |
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375 | (1) |
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375 | (6) |
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375 | (1) |
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376 | (1) |
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376 | (1) |
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377 | (1) |
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378 | (1) |
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378 | (1) |
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379 | (1) |
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379 | (2) |
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The figure of the Earth and of the planets |
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381 | (18) |
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381 | (1) |
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Proposition XVIII and the historical background |
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381 | (3) |
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Proposition XIX: the method of the canals |
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384 | (10) |
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Newton's method of the canals |
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384 | (2) |
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386 | (3) |
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Newton's determination of g(ob)pole/g(ob)eq and ε/m |
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389 | (3) |
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Application to the figure of Jupiter |
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392 | (2) |
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The variation of gravity over an oblate spheroid |
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394 | (5) |
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394 | (2) |
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396 | (3) |
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399 | (20) |
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399 | (2) |
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401 | (2) |
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The tidal force of a distant body acting on the boundary of a spherical body |
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401 | (2) |
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Proposition XXIV: an annotated version |
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403 | (8) |
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Propositions XXV, XXXVI, and XXXVII |
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411 | (4) |
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411 | (1) |
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412 | (1) |
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413 | (1) |
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414 | (1) |
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Appendix: the equilibrium theory of the tides |
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415 | (4) |
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419 | (36) |
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419 | (1) |
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Propositions XVII and XXI |
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420 | (1) |
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420 | (1) |
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420 | (1) |
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Propositions XXII and XXIII |
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421 | (2) |
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421 | (2) |
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423 | (1) |
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423 | (2) |
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425 | (1) |
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425 | (1) |
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426 | (4) |
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The variation in the ascending node, Ω: Proposition XXX |
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430 | (4) |
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430 | (3) |
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433 | (1) |
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434 | (1) |
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434 | (7) |
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436 | (3) |
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Newton's transformation of the equation dΩ/dt for a Kepler ellipse |
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439 | (2) |
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Propositions XXXII and XXXIII |
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441 | (2) |
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441 | (2) |
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443 | (1) |
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The variation of the inclination |
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443 | (5) |
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444 | (1) |
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445 | (1) |
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446 | (1) |
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446 | (2) |
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448 | (7) |
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449 | (1) |
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The motion of the apogee and the `Portsmouth equation' |
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450 | (5) |
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The precession of the equinoxes |
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455 | (22) |
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455 | (1) |
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On the precession of the equinoxes: a current treatment |
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456 | (10) |
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458 | (1) |
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459 | (1) |
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The equations governing precession and nutation |
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459 | (6) |
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The solar contribution to the precession |
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465 | (1) |
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The lunar contribution to the precession and the lunisolar precession |
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465 | (1) |
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Moment of momentum, moment of inertia, and circulation |
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466 | (6) |
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467 | (2) |
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469 | (1) |
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470 | (1) |
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471 | (1) |
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Proposition XXXIX: to find the precession of the equinoxes |
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472 | (3) |
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475 | (2) |
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477 | (60) |
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477 | (1) |
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Lemma IV and Proposition XL |
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478 | (3) |
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478 | (2) |
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480 | (1) |
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480 | (1) |
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Lemma V: Newton's theory of interpolation |
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481 | (19) |
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Newton's treatment in Methodus Differentialis |
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482 | (1) |
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483 | (1) |
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484 | (2) |
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A modern version of Proposition 2 |
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486 | (2) |
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A modern version of Proposition 3 |
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488 | (7) |
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495 | (1) |
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Proposition 3: Cases 1 & 2 |
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495 | (2) |
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497 | (1) |
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498 | (2) |
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500 | (14) |
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500 | (1) |
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501 | (1) |
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502 | (3) |
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505 | (3) |
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508 | (1) |
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Lemma X and `Lambert's theorem' |
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509 | (1) |
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509 | (2) |
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511 | (2) |
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513 | (1) |
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Propositions XLI and XLII |
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514 | (21) |
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514 | (1) |
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|
515 | (1) |
|
The formulation of the problem |
|
|
516 | (1) |
|
Newton's method of solution in the orbital plane |
|
|
517 | (3) |
|
Newton's formulation of the solution |
|
|
520 | (10) |
|
|
530 | (3) |
|
|
533 | (2) |
|
|
535 | (2) |
|
|
537 | (58) |
|
The effect of air-drag on the descent of bodies |
|
|
539 | (16) |
|
Newton's problem and its solution |
|
|
539 | (2) |
|
Lemma III and Proposition XV |
|
|
541 | (8) |
|
|
541 | (2) |
|
|
543 | (3) |
|
|
546 | (1) |
|
|
546 | (1) |
|
|
546 | (1) |
|
|
546 | (1) |
|
|
547 | (1) |
|
|
547 | (1) |
|
|
547 | (2) |
|
|
549 | (1) |
|
|
549 | (2) |
|
|
549 | (1) |
|
|
550 | (1) |
|
|
550 | (1) |
|
|
550 | (1) |
|
|
550 | (1) |
|
|
550 | (1) |
|
|
551 | (1) |
|
An alternative method of solution of Newton's problem |
|
|
551 | (4) |
|
The solid of least resistance |
|
|
555 | (16) |
|
|
555 | (1) |
|
|
556 | (2) |
|
The Scholium, I: the frustum of minimum resistance |
|
|
558 | (1) |
|
The Scholium, II: the solid of least resistance |
|
|
559 | (8) |
|
Newton's manner of solution |
|
|
562 | (5) |
|
|
567 | (4) |
|
|
567 | (1) |
|
The Euler-Lagrange equations |
|
|
567 | (1) |
|
|
568 | (1) |
|
|
569 | (2) |
|
The problem of the brachistochrone |
|
|
571 | (8) |
|
|
571 | (2) |
|
Newton's anonymous solution |
|
|
573 | (2) |
|
The solution derived from its extremal property |
|
|
575 | (4) |
|
Newton's manner of solution |
|
|
577 | (2) |
|
The velocity of sound and of long waves in canals |
|
|
579 | (16) |
|
|
579 | (1) |
|
Propositions XLI--XLIII: Newton's conception of wave propagation |
|
|
580 | (1) |
|
Propositions XLIV--XLVI: the propagation of long waves in canals |
|
|
581 | (5) |
|
|
581 | (1) |
|
|
582 | (1) |
|
|
582 | (1) |
|
|
583 | (1) |
|
|
583 | (1) |
|
|
583 | (1) |
|
|
584 | (1) |
|
|
584 | (1) |
|
The theory of long waves in canals |
|
|
584 | (2) |
|
A standard treatment of the velocity of sound |
|
|
586 | (1) |
|
|
587 | (4) |
|
|
588 | (1) |
|
|
589 | (1) |
|
|
589 | (1) |
|
|
590 | (1) |
|
|
591 | (4) |
Epilogue |
|
595 | |