One work of Aristarchus, _On the sizes and distances of the Sun and Moon_, which is extant in Greek, is highly interesting in itself, though it contains no word of the heliocentric hypothesis. Thoroughly classical in form and style, it lays down certain hypotheses and then deduces therefrom, by rigorous geometry, the sizes and distances of the sun and moon. If the hypotheses had been exact, the results would have been correct too; but Aristarchus in fact assumed a certain angle to be 87° which is really 89° 50', and the angle subtended at the centre of the earth by the diameter of either the sun or the moon to be 2°, whereas we know from Archimedes that Aristarchus himself discovered that the latter angle is only 1/2°. The effect of Aristarchus's geometry is to find arithmetical limits to the values of what are really trigonometrical ratios of certain small angles, namely 1/18 > sin 3° > 1/20, 1/45 > sin 1° > 1/60, 1 > cos 1° > 89/90. The main results obtained are (1) that the diameter of the sun is between 18 and 20 times the diameter of the moon, (2) that the diameter of the moon is between 2/45ths and 1/30th of the distance of the centre of the moon from our eye, and (3) that the diameter of the sun is between 19/3rds and 43/6ths of the diameter of the earth. The book contains a good deal of arithmetical calculation. Archimedes was born about 287 B. C. and was killed at the sack of Syracuse by Marcellus's army in 212 B. C. The stories about him are well known, how he said 'Give me a place to stand on, and I will move the earth' (πα βω και κινω ταν γαν {pa bô kai kinô tan gan}); how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ἑυρηκα, ἑυρηκα {heurêka, heurêka}; and how, the capture of Syracuse having found him intent on a figure drawn on the ground, he said to a Roman soldier who came up, 'Stand away, fellow, from my diagram.' Of his work few people know more than that he invented a tubular screw which is still used for pumping water, and that for a long time he foiled the attacks of the Romans on Syracuse by the mechanical devices and engines which he used against them. But he thought meanly of these things, and his real interest was in pure mathematical speculation; he caused to be engraved on his tomb a representation of a cylinder circumscribing a sphere, with the ratio 3/2 which the cylinder bears to the sphere: from which we infer that he regarded this as his greatest discovery.