where α, β are angles not greater than a right angle and α > β. Aristarchus also, in this proposition, cites 7/5 as an approximation by defect to the value of √2, an approximation found by the Pythagoreans and quoted by Plato. The trigonometrical equivalent of the result obtained in Prop. 7 is
Prop. 8 states that, when the sun is totally eclipsed, the sun and moon are comprehended by one and the same cone which has its vertex at our eye. Aristarchus supports this by the arguments (1) that, if the sun overlapped the moon, it would not be totally eclipsed, and (2) that, if the sun fell short (i.e. was more than covered), it would remain totally eclipsed for some time, which it does not (this, he says, is manifest from observation). It is clear from this reasoning that Aristarchus had not observed the phenomenon of an _annular_ eclipse of the sun; and it is curious that the first mention of an annular eclipse seems to be that quoted by Simplicius from Sosigenes (second century, A.D.), the teacher of Alexander Aphrodisiensis.
It follows (Prop. 9) from Prop. 8 that the diameters of the sun and moon are in the same ratio as their distances from the earth respectively, that is to say (Prop. 7) in a ratio greater than 18 : 1 but less than 20 : 1. Hence (Prop. 10) the volume of the sun is more than 5832 times and less than 8000 times that of the moon.
By the usual geometrical substitute for trigonometry Aristarchus proves in Prop. 11 that the diameter of the moon has to the distance between the centre of the moon and our eye a ratio which is less than 2/45ths but greater than 1/30th. Since the angle subtended by the moon’s diameter at the observer’s eye is assumed to be 2°, this proposition is equivalent to the trigonometrical formula
Having proved in Prop. 4 that, so far as our perception goes, the dividing circle in the moon is indistinguishable from a great circle, Aristarchus goes behind perception and proves in Prop. 12 that the diameter of the dividing circle is less than the diameter of the moon but greater than 89/90ths of it. This is again because half the angle subtended by the moon at the eye is assumed to be 1° or 1/90th of a right angle. The proposition is equivalent to the trigonometrical formula