§ 107. Publication of the _Astronomiae Instauratae Mechanica_ and of the star catalogue: invitation from the Emperor 139 § 108. Life at Benatek: co-operation of Kepler: death 140 § 109. Fate of Tycho’s instruments and observations 141 § 110. Estimate of Tycho’s work: the accuracy of his observations: improvements in the art of observing 141 § 111. Improved values of astronomical constants. Theory of the moon: the _variation_ and the _annual equation_ 143 § 112. The star catalogue: rejection of trepidation: unfinished work on the planets 144 CHAPTER VI. GALILEI (FROM 1564 A.D. TO 1642 A.D.), §§ 113-134 145-178 § 113. Early life 145 § 114. The pendulum 146 § 115. Diversion from medicine to mathematics: his first book 146 § 116. Professorship at Pisa: experiments on falling bodies: protests against the principle of authority 147 § 117. Professorship at Padua: adoption of Coppernican views 148 § 118. =The telescopic discoveries.= Invention of the telescope by _Lippersheim_: its application to astronomy by _Harriot_, _Simon Marius_, and Galilei 149 § 119. The _Sidereus Nuncius_: observations of the moon 150 § 120. New stars: resolution of portions of the Milky Way 151 § 121. The discovery of Jupiter’s _satellites_: their importance for the Coppernican controversy: controversies 151 § 122. Appointment at the Tuscan court 153 § 123. Observations of Saturn. Discovery of the phases of Venus 154 § 124. Observations of _sun-spots_ by Fabricius, Harriot, _Scheiner_, and Galilei: the _Macchie Solari_: proof that the spots were not planets: observations of the _umbra_ and _penumbra_ 154 § 125. Quarrel with Scheiner and the Jesuits: theological controversies: _Letter to the Grand Duchess Christine_ 157 § 126. Visit to Rome. The first condemnation: prohibition of Coppernican books 159 § 127. Method for finding longitude. Controversy on comets: _Il Saggiatore_ 160