equal to 87°] (Heath 1913:353). Proposition 1 states that two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres (Heath 1913:354). Proposition 2 states that if a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere (Heath 1913:358). Proposition 3 states that the circle in the Moon which divides the dark and the bright portions is least when the cone comprehending both the Sun and the Moon has its vertex at our eye (Heath 1913:362). Proposition 4 states that the circle which divides the dark and the bright portions in the Moon is not perceptibly different from a great circle in the Moon (Heath 1913:365). Proposition 6 states that the Moon moves [in an orbit] lower than [that of] the Sun, and, when it is halved, is distant less than a quadrant from the Sun (Heath 1913:372). Proposition 7 states that the distance of the Sun from the Earth is greater than 18 times, but less than 20 times, the distance of the Moon from the Earth (Heath 1913:377). In other words, the Sun is 18 to 20 times farther away and wider than the Moon. Proposition 13 states that the straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the Moon move is less than double of the diameter of the Moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the Sun, but has to it a ratio greater than that which 21 has to 225. But it has to the straight line drawn from the centre of the Sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10 125 (Heath 1913:394). Proposition 14 states that the straight line joined from the centre of