"In his book on sizes and distances Aristarchus lays down these six hypotheses: 1. That the moon receives light from the sun. 2. That the earth is in the relation of a point and centre to the sphere in which the moon moves. 3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye. 4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant. 5. That the breadth of the (earth's) shadow is (that) of two moons. 6. That the moon subtends one fifteenth part of a sign of the zodiac. Now the first, third, and fourth of these hypotheses practically agree with the assumptions of Hipparchus and Ptolemy. For the moon is illuminated by the sun at all times except during an eclipse, when it becomes devoid of light through passing into the shadow which results from the interception of the sun's light by the earth, and which is conical in form; next the (circle) dividing the milk-white portion which owes its colour to the sun shining upon it and the portion which has the ashen colour natural to the moon itself is indistinguishable from a great circle (in the moon) when its positions in relation to the sun cause It to appear halved, at which times (a distance of) very nearly a quadrant on the circle of the zodiac is observed (to separate them); and the said dividing circle is in the direction of our eye, for this plane of the circle if produced will in fact pass through our eye in whatever position the moon is when for the first or second time it appears halved. But, as regards the remaining hypotheses, the aforesaid mathematicians have taken a different view. For according to them the earth has the relation of a point and centre, not to the sphere in which the moon moves, but to the sphere of the fixed stars, the breadth of the (earth's) shadow is not (that) of two moons, nor does the moon's diameter subtend one fifteenth part of a sign of the zodiac, that is, 2°. According to Hipparchus, on the one hand, the circle described by the moon is measured 650 times by the diameter of the moon, while the (earth's) shadow is measured by it 2½ times at its mean distance in the conjunctions; in Ptolemy's view, on the other hand, the moon's diameter subtends, when the moon is at its greatest distance, a circumference of 0° 31' 20", and when at its least distance, of 0° 35' 20", while the diameter of the circular section of the shadow is, when the moon is at its greatest distance, 0° 40' 40", and when the moon is at its least distance, 0° 46'. Hence it is that the authors named have come to different conclusions as regards the ratios both of the distances and of the sizes of the sun and moon."