Dante and the early astronomers

by M. A. Orr

They did indeed represent very beautifully the celestial motions, and also in a general way the variations in brightness or diameter, but their unreality is betrayed by the fact that the latter was often grossly exaggerated, as for instance with the moon, whose epicycle had to be so large, in order to represent her motions, that at perigee she ought to appear twice as large in diameter as at apogee! Ptolemy can hardly have failed to notice this, though he does not mention it. With regard to the planets, he is careful to tell us that he knows no way of finding their real distances: the ratios of their epicycles to their deferents he had carefully computed in each case, but to estimate the diameters in stadia was utterly beyond his powers. He believed, however, that all were very much nearer than the stars, and more distant than the moon, and he found universal agreement among astronomers in placing Saturn, Jupiter, and Mars, beyond the sun, in order of the periods of their deferents, which were all longer than a year. Venus and Mercury, however, had periods the same as the sun: on which side of him, then, must they be placed? Some modern authors, Ptolemy says, thought they were beyond the sun, but he agrees with the most ancient, and places them between sun and moon. The general disposition of the heavenly bodies according to Ptolemy is shown in [Fig. 35], with the periods of epicycles and deferents, and the directions of the several movements, but epicycles and spaces between deferents are all made equal. It will be noted that the sun has no epicycle and that the moon turns on hers in a reverse direction, that the centres of those of Mercury and Venus are in a line with the sun, while the lines joining Mars, Jupiter, and Saturn to the centres of theirs are in each case parallel with the line joining Earth and Sun. Beyond all the wandering stars is the sphere of the fixed stars, moving in its vast period of 36,000 years; and the whole system is carried round Earth in one great revolution of a day and a night.

Fig. 35. The Ptolemaic system.

The arcs of circles are portions of the deferents which carry round the smaller circles (the epicycles) in the periods named at the side.

And here we have to note one astonishing fact. Although the planets were all beyond reach of measurement, it was not so with the moon, which the Greeks rightly recognized as our nearest neighbour. They had actually achieved the long-desired feat of measuring her real size and distance.

The fundamental principle on which they worked is easily explained. If we look out of a window at a tree in the garden, it appears against a background of some distant scenery; if we alter our position by walking to another window, the tree changes its place with regard to the relatively motionless background, and its apparent change of place bears a definite proportion to its distance from us. A tree near the house changes its position greatly, a tree at the far end of the garden much less. From careful measurements of the angle through which the tree appears to have moved and the distance we have walked between the two windows, the distance of the tree from the house may be easily deduced.

The tree in the garden is the moon, the distant landscape is the star-spangled sky. The space between the two windows must be increased to thousands of miles, but the astronomer need not walk it. If he makes his first observation when the moon has just risen, carefully measuring her position among the stars, the revolving Earth will carry him to a new position in a few hours, when with the moon high in the sky he can once more compare her position with the same stars, and from the change he then finds he can deduce her distance. The Greeks thought it was the sky, and not the earth which moved, but this makes no difference, as the question is one of relative motion only.

The problem, however, when applied to the moon, is a complicated one, implying not only skill in trigonometry and the possession of suitable instruments for measuring the necessary angles, but also accurate knowledge of the size of the earth and the motions of the moon, since her progress eastward among the stars during the interval between the two observations must be allowed for. Hipparchus and the astronomers of Alexandria were the first to qualify themselves for attacking this difficult problem, and the proof of their success is that Ptolemy’s value for the distance of the moon is very near the truth as obtained by modern methods. The same method is now used, only it is found better to do what was not possible for the Greeks, namely to compare observations made at places several thousands of miles distant, for instance Greenwich and the Cape, instead of allowing the same place to be moved by the earth’s rotation.

When the distance of the moon had been thus discovered, it was a very simple matter to find her real size from her apparent size.