Ptolemy's hypotheses of astronomy

Near the beginning of Book I of the Almagest, Ptolemy sets out the hypotheses of Ptolemaic astronomy:-

On the Order of the Theorems

2. The work that we have projected commences with a consideration of the general relation between the earth as a whole and the heavens as a whole. Of the special treatments that follow, the first part deals with the position of the ecliptic, the places inhabited by the human race, and the differences among the successive places, in each separate horizon, along the curvature of the earth's surface. The preliminary study of these relations makes easier the examination of the subsequent questions. The second part gives an account of the motion of the sun and the moon and of the phenomena that depend on these motions. For without the previous understanding of these matters it would be impossible to set forth a complete theory of the stars. Since the theory of the stars is contained, in accordance with the general plan, in the concluding portion of this essay, the investigation of the sphere of the so-called fixed stars would properly find its place there, and the material on the five so-called planets would follow. We shall try to set forth all this material using as the basic foundations for knowledge the manifest phenomena themselves and those recorded observations of the ancients and the moderns about which there is no dispute; and we shall seek to fit the propositions together by geometrical proofs.

With respect to the general portion of the treatise the following preliminary assumptions are to be made: (1) that the heaven is spherical in form and rotates as a sphere; (2) that the earth, too, viewed as a complete whole, is spherical in form; (3) that it is situated in the middle of the whole heaven, like a centre; (4) that by reason of its size and its distance from the sphere of fixed stars the earth bears to this sphere the relation of a point; (5) that the earth does not participate in any locomotion. We shall say a few words by way of commentary on each of these propositions.

That the Heaven Rotates as a Sphere

3. It is reasonable to assume that the first ideas on these matters came to the ancients from observation such as the following. They saw the sun and the moon and the other stars moving from east to west in circles always parallel to each other; they saw the bodies begin to rise from below, as if from the earth itself, and gradually to rise to their highest point, and then, with a correspondingly gradual decline, to trace a downward course until they finally disappeared, apparently sinking into the earth. And then they saw these stars, once more, after remaining invisible for a time, make a fresh start and in rising and setting repeat the same periods of time and the same places of rising and setting with regularity and virtual similarity.

They were, however, led to the view of a spherical heaven chiefly by the observed circular motion described about one and the same centre by those stars that are always above the horizon. For this point was, necessarily, the pole of the heavenly sphere, since the stars that are nearer this pole revolve in smaller circles, whereas those further away make larger circles, proportionately to their distance, until the distance reaches that of the stars not always visible. And of these latter they observed that those stars nearer the stars that are always visible remained invisible for a shorter time while those further away remained invisible for a correspondingly longer time. And so, from these phenomena alone they first conceived the aforesaid idea, and then from the consideration of its consequences they adopted the other ideas that follow from it, since all the phenomena without qualification refuted the alternative hypotheses.

For example, if one should suppose, as some have, that the motion of the stars proceeds by a straight line without limit, how could one explain the fact that the daily motion of each star is always seen to begin from the same point? How could the stars in their unlimited motion turn back? And if they did turn back, how could this escape observation? Or how could they fail eventually to become altogether invisible, since they would appear ever smaller and smaller? In point of fact, however, they appear larger when near the region where they disappear, and are only gradually occulted and, as it were, cut off by the surface of the earth.

Again, the suggestion that the stars are kindled when they rise from the earth and again are snuffed out when they return to the earth is quite contrary to reason. For even if one should grant that the arrangement, size, and number of the stars, and their distances and intervals in space and time could have been the fulfilment of mere random and accidental procedure and that one part of the earth (the eastern part) had throughout it a kindling force, while the other (the western part) had an extinguishing force, or rather that the same part acted as a kindler from the point of view of some and as an extinguisher from the point of view of others, and that of the stars the very same ones were already kindled or extinguished, as the case might be, for some observers, but not yet for others - if, I repeat, one should grant all this, absurd as it is, what of the stars always visible, those that neither rise nor set? Why should the stars that are kindled and extinguished not rise and set everywhere? Why should those not subject to such kindling and extinguishing always be above the horizon in all latitudes? For surely the stars that for some observers are always kindled and extinguished cannot be the same as those that for other observers are never kindled and extinguished. (Yet the proponents of the hypothesis of kindling would have to assume that they are the same) for it is quite evident that the same stars rise and set for some observers (i.e., those further south) whereas they neither rise nor set for others (i.e., those further north).

In a word, if one should suppose any other form of motion of the heavens save the spherical, the distances from the earth to the heavenly bodies would necessarily be unequal, however and wherever the earth itself might be supposed to lie situate. Consequently the sizes of the stars and their distances from one another would have to appear unequal to the same observers at each return, since the distances from the observers would sometimes be greater and at other times smaller. But this is not seen to be the case. For what makes the apparent size of a heavenly body greater when it is near the horizon is not its smaller distance but the vaporous moisture surrounding the earth between our eye and the heavenly body. It is the same as when objects immersed in water appear larger, and in fact the more deeply immersed the larger.

The hypothesis of spherical motion finds support also in the fact that on any other hypothesis save this one alone it is impossible that the instruments for measuring hours should be correct. There is also support in the following fact. Just as the motion of the heavenly bodies is completely without hindrance and the smoothest of all motions, and the most easily moved of all shapes is the circular for plane figures and the spherical for solids, so also since the polygon with the greater number of sides is the larger of regular polygons having equal perimeters, it follows that in the case of plane figures the circle is greater than any polygon of equal perimeter, and in the case of solid figures the sphere is greater. And the heaven is greater than all other bodies.

Various physical considerations, too, lead to the same conclusion.

Thus the aether consists of finer and more homogeneous parts than does any other body. Now surfaces of bodies of homogeneous parts are themselves of homogeneous parts, and the circular surface in the case of plane figures and the spherical surface in the case of solid figures are the only surfaces that consist of homogeneous parts. The aether not being a plane surface but a solid may therefore be inferred to be of spherical form. A similar inference may be made from the fact that nature has constructed all earthly and destructible bodies entirely of circular forms but forms not having homogeneous parts, while she has constructed the divine bodies in the aether of spherical form having homogeneous parts. For if these bodies were flat or quoit-shaped, their form would not appear circular to all observers at the same time from different places of the earth. Hence it is reasonable to infer that the aether that encloses the heavenly bodies, being of the same nature, is of spherical form, and, because of its composition out of homogeneous parts, moves with uniform circular motion.

The Absolute Immobility of the Earth

In the same way as before it can be proved that the earth cannot make any movement whatever in the aforesaid oblique direction, or ever change its position at all from its place at the centre; for the same results would, in that case, have followed as if it had happened to be placed elsewhere than at the centre. So I, for one, think it is gratuitous for any one to inquire into the causes of the motion towards the centre when once the fact that the earth occupies the middle place in the universe, and that all weights move towards it, is made so patent by the observed phenomena themselves. The ground for this conviction which is readiest to hand, seeing that the earth has been proved to be spherical and situated in the middle of the universe, is this simple fact: in all parts of the earth without exception the tendencies and the motions of bodies which have weight - I mean their own proper motions - always and everywhere operate at right angles to the (tangent) plane drawn evenly through the point of contact where the object falls. That this is so makes it also clear that, if the objects were not stopped by the surface of the earth, they would absolutely reach the centre itself, since the straight line leading to the centre is always at right angles to the tangent-plane to the sphere drawn through the intersection at the point of contact.

All who think it strange that such an immense mass as that of the earth should neither move itself nor be carried somewhere seem to me to look to their own personal experience, and not to the special character of the universe, and to go wrong through regarding the two things as analogous. They would not, I fancy, think the fact in question to be strange if they could realize that the earth, great as it is, is nevertheless, when compared with the enclosing body, in the relation of a point to that body. For in this way it will seem to be quite possible that a body relatively so small should be dominated and pressed upon with equal and similarly directed force on all sides by the absolutely greatest body formed of like constituents, there being no up and down in the universe any more than one would think of such things in an ordinary sphere. So far as the composite objects in the universe, and their motion on their own account and in their own nature are concerned, those objects which are light, being composed of fine particles, fly towards the outside, that is, towards the circumference, though their impulse seems to be towards what is for individuals "up," because with all of us what is over our heads, and is also called "up," points towards the bounding surface; but all things which are heavy, being composed of denser particles, are carried towards the middle, that is, to the centre, though they seem to fall "down," because, again, with all of us the place at our feet, called "down," itself points towards the centre of the earth, and they naturally settle in a position about the centre, under the action of mutual resistance and pressure which is equal and similar from all directions. Thus it is easy to conceive that the whole solid mass of the earth is of huge size in comparison with the things that are carried down to it, and that the earth remains unaffected by the impact of the quite small weights (falling on it), seeing that these fall from all sides alike, and the earth welcomes, as it were, what falls and joins it. But, of course, if as a whole it had had a common motion, one and the same with that of the weights, it would, as it was carried down, have got ahead of every other falling body, in virtue of its enormous excess of size, and the animals and all separate weights would have been left behind floating on the air, while the earth, for its part, at its great speed, would have fallen completely out of the universe itself. But indeed this sort of suggestion has only to be thought of, in order to be seen to be utterly ridiculous.

Certain thinkers, though they have nothing to oppose to the above arguments, have concocted a scheme which they consider more acceptable, and they think that no evidence can be brought against them if they suggest for the sake of argument that the heaven is motionless, but that the earth rotates about one and the same axis from west to east, completing one revolution approximately every day, or alternatively that both the heaven and the earth have a rotation of a certain amount, whatever it is, about the same axis, as we said, but such as to maintain their relative situations.

These persons forget however that, while, so far as appearances in the stellar world are concerned, there might, perhaps, be no objection to this theory in the simpler form, yet, to judge by the conditions affecting ourselves and those in the air about us, such a hypothesis must be seen to be quite ridiculous. Suppose we could concede to them such an unnatural thing as that the most rarefied and lightest things either do not move at all or do not move differently from those of the opposite character - when it is clear as day that things in the air and less rarefied have swifter motions than any bodies of more earthy character - and that (we could further concede that) the densest and heaviest things could have a movement of their own so swift and uniform - when earthy bodies admittedly sometimes do not readily respond even to motion communicated to them by other things - yet they must admit that the rotation of the earth would be more violent than any whatever of the movements which take place about it, if it made in such a short time such a colossal turn back to the same position again, that everything not actually standing on the earth must have seemed to make one and the same movement always in the contrary sense to the earth, and clouds and any of the things that fly or can be thrown could never be seen travelling towards the east, because the earth would always be anticipating them all and forestalling their motion towards the east, insomuch that everything else would seem to recede towards the west and the parts which the earth would be leaving behind it.

For, even if they should maintain that the air is carried round with the earth in the same way and at the same speed, nevertheless the solid bodies in it would always have appeared to be left behind in the motion of the earth and air together, or, even if the solid bodies themselves were, so to speak, attached to the air and carried round with it, they could no longer have appeared either to move forwards or to be left behind, but would always have seemed to stand still, and never, even when flying or being thrown, to make any excursion or change their position, although we so clearly see all these things happening, just as if no slowness or swiftness whatever accrued to them in consequence of the earth not being stationary.

JOC/EFR August 2006

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