the Earth to the centre of the Moon has to the straight line cut off from the axis towards the centre of the Moon by the straight line subtending the [circumference] within the Earth's shadow a ratio greater than that which 675 has to 1 (Heath 1913:400). Proposition 15 states that the diameter of the Sun has to the diameter of the Earth a ratio greater than 19/3, but less than 43/6 (Heath 1913:403). This means that the Sun is (a mean of) 6¾ times wider than the Earth, or that the Sun is 13½ Earth-radii wide. The Moon and Sun must then be 20¼ and 387 Earth-radii away from us in order to subtend an angular size of 2º. Proposition 17a in al-Tusi's medieval Arabic version of the book On Sizes states that the ratio of the distance of the vertex of the shadow cone from the center of the Moon (when the Moon is on the axis [that is, at the middle of an eclipse] of the cone containing the Earth and the Sun) to the distance of the center of the Moon from the center of the Earth is greater than the ratio 71 to 37 and less than the ratio 3 to one (Berggren & Sidoli 2007:218). In other words, that the tip of the Earth's shadow cone is between 108/37 and four times farther away than the Moon. Known copies Library of Congress Vatican Exhibit. See also Aristarchus of Samos Eratosthenes (), a Greek mathematician who calculated the circumference of the Earth and also the distance from the Earth to the Sun. Hipparchus (), a Greek mathematician who measured the radii of the Sun and the Moon as well as their distances from the Earth. On the Sizes and Distances (Hipparchus) Posidonius (), a Greek astronomer and mathematician who calculated the circumference of the Earth. Notes Bibliography This was later reprinted, see (). van Helden, A. Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley. Chicago: Univ. of Chicago Pr., 1985. . Ancient Greek astronomical works Astronomy books