The fourth hypothesis runs: ‘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 (i.e. is 87°).’ From this is directly deduced (Hypothesis 6 is not here used) Prop. 7, an elaborate proof that ‘the distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth’, quoted by Plutarch in c. 10. The fact assumed does not appear to be open to observation; perhaps Aristarchus, or a predecessor, arrived at it by comparing the average times taken by the moon over the first and second quarters of her orbit. The true (theoretical) figure is 89° 50´. The sequel is very interesting. Hipparchus, a century later, adopted the result in calculating the parallax of the sun, which he found to be 3´ of arc (more than twenty times too much). This was adopted by Ptolemy in the second century A.D., and remained the official estimate until nearly A.D. 1700, though both Hipparchus and Kepler had protested, the latter stating as his opinion that the parallax could not be greater than one minute of arc, or the distance less than twelve millions of miles. Shortly before A.D. 1700 improved knowledge of the orbit and distances of Mars enabled the sun’s parallax to be reduced to 9-1/2 seconds of arc, and his distance stated at eighty-seven millions of miles, which is not very inadequate. It was a great achievement of Aristarchus, though he led the world into error, to state a reasoned figure at all, and to think in such mighty units. His cosmical speculation is even more daring. It is known to us from this Dialogue (c. 6) and also from Archimedes, who records it in his (extant) _Arenarius_ without comment. Aristarchus proposed to ‘disturb the hearth of the universe’ by his hypothesis that the heaven of the stars is fixed, while the earth has a daily motion on her axis and an annual motion round the sun. It was a brilliant intuition, possible in an age of comparatively simple knowledge, which could not easily have been advanced when the complexity of the several orbits was increasingly realized (see Dreyer, pp. 147-8). Dr. Dreyer (p. 145) makes the interesting suggestion that Aristarchus took the idea from some early form of the system of ‘movable eccentrics’, and, further (p. 157), that if that system had prevailed against that of epicycles, it must have flashed, sooner or later, upon some bright mind, that there was one eccentric point, namely, one in the sun, central to the orbits of all the planets.