When the next type of calculation was possibly used for the first time? If 50% i.e. upper half of Moon were bright would the Sun-Earth to Moon-Earth distances ratio be calculated from just astronomicaly measuring the angle a if e.g. angle a' is set to 35° and if the Sun and Moon are viewed from opposite directions?I assume for a equal to 55° the Sun would be infinitely distant from Earth.
In the third century BCE Aristarchus of Samos estimated the ratio of the distance of the sun to the distance of the moon by observing the angle between the sun and the moon when the moon was exactly half full.