Aristarchus On The Sizes and Distances - Illustrations

Illustrations

Some interactive illustrations of the propositions in On Sizes can be found here:

• Hypothesis 4 states 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 (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 lower than 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 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 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 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 4 times farther away than the moon.