From Kepler to Ptolemy 15

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The Planetary Hypotheses

In the Planetary Hypotheses, Ptolemy lays out his cosmology: that is, the structure and arrangement of the universe. This work answers the question, did Ptolemy believe in the physical truth of the Almagest’s celestial geometry?—with an unambiguous Yes. Contrary to an opinion often expressed by earlier historians, he did not regard it just as a calculational scheme for predicting planetary positions.

In 1454, Georg von Peurbach wrote Theoricae Novae Planetarum (New Theory of the Planets), published posthumously by his successor Regiomontanus in 1472. (There is an English translation by Aiton). It orginated as lecture notes, and along with the Epytoma Joanis de mote regio in almagestu ptolomei (Epitome of Ptolemy’s Almagest; started by Peurbach and completed by his pupil Regiomontanus) became the two main astronomy textbooks for the next 150 years. It seems likely that Peurbach’s work derived from the Planetary Hypotheses, probably through the Arabic intermediary Ibn al-Haytham.

Only Book I of the Planetary Hypotheses survives in the original Greek, but there is an extant Arabic translation of the whole thing (Books I&II)¹.

It is a late work, perhaps Ptolemy’s last. We’ve seen already in post 13 how he took the opportunity to correct his earlier work. He says:

We shall make the exposition, so far as the general assumptions are concerned, in agreement with the things delineated in the Syntaxis [i.e., the Almagest], so far as the details are concerned, following the corrections we have produced in many places on the basis of more continuous observations, either corrections to the models themselves, or corrections to the spatial ratios, or corrections to the periods of restitutions.

—Hamm (Bk I)

Hamm argues that ‘models’ is a better translation of the Greek ‘υπoθέσισ than ‘hypothesis’. The Planetary Hypotheses contemplates both geometrical and mechanical models—actual physical devices, maybe with gears. Book I begins:

We have worked out, Syrus², the models of heavenly motions through the books of the Mathematical Syntaxis [Almagest]… Here we have taken on the task to set out the thing itself briefly, so that it can be more readily comprehended by both ourselves and by those choosing to arrange the models in an instrument…

Murschel remarks, “the models themselves are clearly meant to be manufactured”, although (as she later qualifies) “his instruction manual… poses far too great a challenge for the instrument maker.”

After these preliminaries, Book I summarizes the geometrical models in about 20 pages (in Hamm’s translation). Only the latitude theory and the parameter values see changes from the Almagest.

Next he computes the size of the universe, based on three assumptions:

1. The order of the heavenly bodies is Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
2. The ratio of the least to greatest distance from the Earth for each body is given by the models of the Almagest.
3. Bounding spheres with these radii are nested without gaps.

He gets a partial confirmation in that the maximum distance of Venus works out to 1079 Earth radii, while the minimum distance to the Sun had been found independently as 1160 Earth radii³. He suggests that perhaps the actual minimum solar distance is the lower figure. His final figure for the outer bounding sphere of Saturn: 19865 Earth radii, or roughly 0.8AU in modern units. The correct figure is about 10.5AU.

Summing up, he writes:

If (the universe is constructed) according to our description of it, there is no space between the greatest and least distances (of adjacent spheres), and the sizes of the surfaces that separate one sphere from another do not differ from the amounts we mentioned. This arrangement is most plausible, for it is not conceivable that there be in Nature a vacuum, or any meaningless and useless thing. The distances of the spheres that we have mentioned are in agreement with our hypotheses. But if there is space or emptiness between the (spheres), then it is clear that the distances cannot be smaller, at any rate, than those mentioned. [trans. by Goldstein]

Book I finishes off with estimates of the sizes of the Sun, Moon, and planets, and some other stuff I won’t go into. (Look up arcus visionis and ‘acronychal risings’ if you’re really curious.)

Book II presents Ptolemy’s physical models—his cosmology, also called the Ptolemaic system. People often describe this as a hybrid of Ptolemy’s celestial geometry with Aristotle’s cosmology. I won’t go into the latter in detail; as mentioned already, it consists of 56 homocentric spheres. The outermost sphere of fixed stars rotates diurnally, carrying the inner spheres with it, which is what you want. However, you don’t want Saturn’s sphere to transmit its rotation to Jupiter’s (for example), so Aristotle introduced counter-rotating spheres to cancel out the motion of enclosing spheres. The Aristotelian system failed utterly when it came to quantitative predictions.

Ptolemy’s cosmology differs from Aristotle’s in several ways, apart from its use of the deferent-epicycle-equant geometry. Ptolemy’s system is less crassly mechanical than Aristotle’s: the shells are made of aether! Earthly analogs are inappropriate. No attached poles, for example.

Murschel (p.39) summarizes the main features of Ptolemy’s celestial physics:

In order to surpass these older cosmological systems, Ptolemy nevertheless turns to another common metaphysical concept, the soul. He believes that “the planets are ensouled (mutanaffasa) and are moved with a voluntary motion”. It is this soul that maintains the faculty for producing brightness and motion.…

Each planet, Ptolemy explains, has the power to direct its own motion and the motions of the adjacent celestial bodies within its own system…the planet’s psychic faculty sends motive emissions to the epicycle, then to the deferent, then to the outermost of that planet’s celestial bodies, which is concentric with the Earth. …Moreover, Ptolemy contends, the motive emission will have no difficulty in passing from one celestial body to the next, just as that other emission which we call brightness easily travels through all of the celestial bodies as well as the elements of our own atmosphere.

—Murschel (p.39)

Ptolemy is more economical than Aristotle. He eliminates most of the counter-rotating shells. As a further economy, he suggests replacing complete spherical shells with “sawn-off pieces”: spherical zones, extending a bit above and below the sphere’s equator. His geometrical models don’t need most of the sphere. The Arabic for “sawn-off piece” is manshūrāt; indeed, the Arabic title for the Planetary Hypotheses is The book of the sawn-off pieces. These manshūrāt can be hollow, like a bracelet, or solid, like a tambourine⁴.

Time to look at Ptolemy’s physical models. Ptolemy offers versions with both complete shells and with sawn-off pieces. The figure above shows a cross-section in the ecliptic plane of the full-shell version for all the planets except Mercury. The planet is embedded in a solid epicycle shell that rotates uniformly. This is embedded in the shell of the deferent, just thick enough to contain it. The deferent and the epicycle shells have different axes of rotation, with the epicycle’s axis perpendicular to the plane of the ecliptic for all the outer planets. The deferent’s axis is tilted with respect to the ecliptic to account for the planet’s varying latitude. The deferent shell rotates at a variable rate, governed by the equant.

The epicycle region actually consists of two shells, a solid inside a hollow. Without the hollow shell, the deferent would impart its rotation to the solid shell. As it is, the deferent merely carries the center of the epicycle around in a circle.

The deferent shell lies between two spheres, each concentric with the center of the Earth. The deferent shell is off-center, reflecting the eccentricity of the deferent circle in the geometric model. The inner and outer spheres mark the least and greatest distances of the planet from the Earth. They belong to the family of nested spheres used to calculate the size of the universe in Book I. Thus the outer sphere for Jupiter fits snugly inside the inner sphere of Saturn, and likewise for the rest of the planets.

The regions marked parecliptic in the figure rotate very slowly about an axis perpendicular to the ecliptic; the rotation rate matches the precession of the equinoxes.

Altogether then we have five celestial bodies for each planet except Mercury: two paraecliptic regions, the deferent shell, and two shells for the epicycle. Mercury requires seven bodies. The Moon requires six, and the Sun three.

Outside all the planets we have the spherical shell of fixed stars, also rotating to match the precession of the equinoxes. Outside that we have a sphere that rotates diunally around an axis perpendicular to the celestial equator, and not the ecliptic. This outermost shell imparts the diurnal rotation to everything inside it.

This figure shows a cross-section of the scheme with sawn-off pieces, manshūrāt:

The outer manshūrāt is the two parecliptic regions now fused into one, and just high enough to accomodate the inner tilted manshūrāt of the deferent.

Three complete shells remain in this approach: the outermost sphere, the sphere of fixed stars, and “loose aether”. The loose aether fills up all the space left vacant by the manshūrāts. Ptolemy, like Aristotle, abhors a vacuum.

Ptolemy gives two “celestial body counts”, one with complete shells, one with manshūrāts. The first count includes “movers” to provide diurnal rotation; they rotate just like the outermost sphere. Here is the complete shell count:

1	outermost sphere
1	fixed stars
7	movers
20	planets except Mercury
7	Mercury
1	Sun
4	Moon
41	Total

It’s not clear why Ptolemy has only one body for the Sun instead of 3. For similar reasons, the Moon should have 6 bodies and not 4. Later on Ptolemy suggests that the outermost sphere can provide diurnal rotation for the whole system, and discards the movers. This would reduce the total count to 34, though it should really be 38 because of the Sun and the Moon.

With the manshūrāt models, his accounting runs:

1	outermost sphere
1	fixed stars
1	loose aether
16	planets except Mercury
5	Mercury
1	Sun
4	Moon
29	Total

Four bodies for the Moon is now correct, while the Sun should be two. So the correct total is now 30. Although the seven “movers” have been left out this time, Ptolemy considers subtracting them again, bringing the total down to 22 (Murschel (p.52), Evans (p.391)).

Ptolemy exhibits different attitudes towards the models of the Almagest and of the Planetary Hypotheses. The Almagest models were the fruit of geometry applied to voluminous data. The Planetary Hypotheses is more speculative.

In the preface to the Almagest, Ptolemy divides theoretical philosophy into three branches: theology, physics, and mathematics. Of these, only mathematics provides “unshakeable knowledge”:

From all this we concluded: that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakeable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry.

—Toomer (p.36)

Pedersen suggests that Ptolemy regarded astronomy as part of mathematics, or at least the astronomy in the Almagest:

It is not said in so many words, but it is as if he thinks as follows: If one studies the changing and corruptible material world the resulting science belongs to physics, even if it has a mathematical form. On the other hand, a study of the unchanging and eternal heavens of a similar mathematical form belongs to the science of mathematics.

—Pedersen (p.30)

Ptolemy broke out two other aspects of astronomy into separate works: the physical structure in the Planetary Hypotheses, and astrology into the Tetrabiblos. Of the latter, Pedersen remarks:

The status of astronomy as a part of mathematics is also, perhaps, the explanation of the very remarkable fact that the Almagest is completely free of astrology. This is not because Ptolemy did not believe in the possibility of making predictions from the stars. On the contrary, his Tetrabiblos is one of the most comprehensive manuals of judicial astrology ever written…. But astrology is concerned with the influence of the celestial world upon the terrestrial, and the influence of the stars is very closely connected with their physical nature.

—Pedersen (p.30)

In at least four places in the Planetary Hypotheses, Ptolemy acknowledges the impossibility of reaching definitive conclusions.

1. Are complete shells needed, or just sawn-off pieces?
2. What is the order of the planets?
3. Are “movers” needed for each planet and the Sun and the Moon, or is the outermost sphere sufficient?
4. Are the spheres tightly nested, or could there be extra space between them?

(See the discussion in Hamm (pp.31–43) for the first two.) In each case Ptolemy states a clear preference, based on simplicity, economy, and/or naturalness. (1) He argues that sawn-off pieces are more economical than full shells. (2) The order of the planets cannot be known with certainty:

But with respect to the Sun, there are three possibilities; either all five planetary spheres lie above the sphere of the Sun just as they all lie above the sphere of the Moon; they all lie below the sphere of the Sun; or some lie above, and some below the sphere of the Sun.

—Goldstein (p.6)

He opts for the third choice as the most natural (the usual dichotomy of inner and outer planets). (3) One can dispense with “movers”, except for the outermost sphere, given certain assumptions about celestial physics (Murschel), and he prefers this, again on grounds of economy. But he includes the movers in his first “body count”. (4) To repeat something quoted above: “But if there is space or emptiness between the (spheres), then it is clear that the distances cannot be smaller, at any rate, than those mentioned.” (Goldstein, p.8)

Ptolemy had good instincts. As we’ve seen, the geometry of the Almagest survives, transposed, as a decent approximation to Keplerian (and hence modern) astronomy. The same cannot be said for any part of the Planetary Hypotheses.

[1] Murschel and Hamm provide full accounts of the complicated history of the textual sources.

[2] Hamm: “Ptolemy dedicated several of his works to Syrus, including the Almagest. Syrus was probably a benefactor and unfortunately this is all that is known about him.”

[3] The actual figure is approximately 23,000 Earth radii. People have gone over Ptolemy’s arithmetic and found minor mistakes. See Hamm (pp.196–197).

[4] That’s the word Evans, Murschel, and Hamm use, quoting (I believe) the Arabic. Tambourines in my experience are not solid.

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