### Parallax Cases

The issue of the negligibility or otherwise of what is discounted takes us to some more complicated cases, such as the various assumptions made by different astronomers in different contexts about parallax. We can distinguish between three main types of case. First, there is the assumption that in relation to the sphere of the fixed stars, the earth may be treated as a point: it is of negligible size, and so it does not matter, in this context, that an observer is not at the centre of the earth, but on its surface, at some distance from that centre.^{[79]} This assumption is set out in Euclid's *Phaenomena* , for instance, and in the elementary textbook of Cleomedes.^{[80]} It also figures in *Syntaxis* book 1 chapter 6, where, moreover, Ptolemy offers a particularly clear statement of the grounds to justify it: the fact that the configurations of the constellations remain unchanged from whatever point on the earth they are observed *indicates* the very great distance of the stars.^{[81]}

A second and far more controversial assumption is that made in

Aristarchus' heliocentric theory, as reported by Archimedes, namely, that not just the earth but the circle in which the earth moves around the sun is as a point in relation to the sphere of the fixed stars (see Figure 2).^{[82]} Archimedes' own comment is that that is, strictly speaking, impossible, since a point has zero magnitude and the fixed stars would then be at infinite distance^{[83]} (a similar point applies, of course, to the first type of parallax case as well). What Aristarchus needs is not that the stars be infinitely, only that they be indefinitely, far away.

The interesting feature is that he evidently incorporated this *into his assumptions* , in part in order to meet a possible objection to heliocentricity. If the earth moves in a circle around the sun (rather than the sun around the earth) there should be, one might think, observable differences in the shapes of the constellations as viewed from different points in the earth's orbit—from the points representing the spring and autumn equinoxes, for instance, at opposite ends of the same diameter of the orbit. Yet no such variation was observed; indeed, stellar parallax was not confirmed until well into the nineteenth century, with the work of Bessel and others around 1835. Aristarchus seems to have appreciated that this otherwise very damaging objection to heliocentricity was no objection at all *provided* that the stars are sufficiently far away. If the diameter of the earth's orbit around the sun is negligible in comparison to the diameter of the sphere of the fixed stars, then you would not expect * observable* variations in the relative positions of the stars, certainly not within the limits of ancient techniques of observation. Unlike Ptolemy's discussion of the size of the earth in *Syntaxis* 1.6, the assumption in the form adopted by Aristarchus was not itself justified by reference to independently observable phenomena; there was no way in which it could be. Rather, this reveals precisely what has to be accepted *among the assumptions* in order for an apparent objection from the side of the phenomena *not* to be the objection it seems. No doubt Aristarchus could have argued that the

inability to confirm an assumption directly does not make it untrue—and Copernicus would have said the same.^{[84]}

The third type of case again comes from Aristarchus, this time from the extant treatise *On the Sizes and Distances of the Sun and Moon* . The second hypothesis set out there is that the earth is as a point, not in relation to the sphere of the fixed stars, but in relation to the *moon's orbit* .^{[85]} In this form the assumption involves discounting *lunar* paral-

lax—as if the position of the observer on the surface of the earth makes no difference to observations of the moon. Yet of course it does. The contrast with the careful and complex discussions in the *Syntaxis* where Ptolemy attempts to determine the allowance that has to be made for lunar parallax^{[86]} is striking and points up the difficulty that the second hypothesis in Aristarchus' treatise presents, indeed, its complete unacceptability *if* we are concerned with trying to establish the *actual* size of the moon and its *actual* distance from the earth.

Yet we should not be misled by one possible way of taking the title of Aristarchus' treatise (*On the Sizes and Distances* . . . ) into thinking that that *was* his aim. Both the hypotheses and the results militate

against such a view. First, his results all take the form of ratios or proportions, giving upper and lower limits for the relative sizes and distances on the basis of the assumptions as set out; no absolute figures are presented.^{[87]} Then, the hypotheses include several that Aristarchus undoubtedly knew to be well wide of the mark. That appears to be the most likely explanation of the notorious sixth hypothesis, that the moon subtends an angle of 2° to the eye;^{[88]} where 1/2° was the usual ancient approximation and is indeed the figure we can attribute to Aristarchus on the basis of a report in Archimedes.^{[89]} Again, the fourth hypothesis simply assumes, with no attempt at justification, that the moon is at 87° to the sun when it appears to be halved.^{[90]} Again, the

fifth takes it that the breadth of the earth's shadow, viz., at the moon, is two moons.^{[91]} Moreover, the second hypothesis itself not only discounts lunar parallax but takes the moon to move in a simple circle with the earth as centre—and no serious Greek astronomer had thought *that* since before Eudoxus.

Such a set of hypotheses would, of course, be an unmitigated disaster in any attempt to arrive at concrete determinations of the actual sizes and distances of the moon and sun. What Aristarchus is doing, rather, is exploring the geometry of the problems. Given certain assumptions—and it will not matter, for the sake of the geometry, that some of the values are a little, and others wildly, inaccurate—what follows? The study is certainly * relevant to* astronomy, in particular because it shows * how* one *could* obtain *actual* solutions to the astronomical parameters, that is, it offers one set of answers to the question of the premises, or data, needed in order to arrive at such solutions. Yet it remains itself essentially a study of the geometry of the problems.^{[92]}