Hypothesis 4. If S be the sun, M the moon and E the earth, the triangle SME is, at the moment when the moon appears to us halved, right-angled at M; and the hypothesis states that the angle at E in this triangle is 87°, or, in other words, the angle MSE, that is, the angle subtended at the sun by the line joining M to E, is 3°. These estimates are decidedly inaccurate, for the true value of the angle MES is 89° 50′, and that of the angle MSE is therefore 10′. There is nothing to show how Aristarchus came to estimate the angle MSE at 3°, and none of his successors seem to have made any direct estimate of the size of the angle.
The assumption in Hypothesis 5 was improved upon later. Hipparchus made the ratio of the diameter of the circle of the earth’s shadow to the diameter of the moon to be, not 2, but 2½ at the moon’s mean distance at the conjunctions; Ptolemy made it, at the moon’s greatest distance, to be inappreciably less than 2⅗.
The sixth hypothesis states that the diameter of the moon subtends at our eye an angle which is 1/15th of 30°, i.e. 2°, whereas Archimedes, as we have seen, tells us that Aristarchus found the angle subtended by the diameter of the sun to be ½° (Archimedes in the same tract describes a rough instrument by means of which he himself found that the diameter of the sun subtended an angle less than 1/164th, but greater than 1/200th of a right angle). Even the Babylonians had, many centuries before, arrived at 1° as the apparent angular diameter of the sun. It is not clear why Aristarchus took a value so inaccurate as 2°. It has been suggested that he merely intended to give a specimen of the calculations which would have to be made on the basis of more exact experimental observations, and to show that, for the solution of the problem, one of the data could be chosen almost arbitrarily, by which proceeding he secured himself against certain objections which might have been raised. Perhaps this is too ingenious, and it may be that, in view of the difficulty of working out the geometry if the two angles in question are very small, he took 3° and 2° as being the smallest with which he could conveniently deal. Certain it is that the _method_ of Aristarchus is perfectly correct and, if he could have substituted the true values (as we know them to-day) for the inaccurate values which he assumes, and could have carried far enough his geometrical substitute for trigonometry, he would have obtained close limits for the true sizes and distances.
