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Ptolemy,
Almagest (ca. 150 A.D.)
Assignments
Ptolemy and Galen each wrote in Greek, and worked in the 2nd century AD during
the period of the Roman empire. Take your time with them; they represent, respectively,
the culmination of ancient mathematical astronomy and ancient anatomy and medicine.
- Review the discussion of Ptolemy's astronomy in Lindberg, Beginnings
of Western Science, Chapter 5, pp. 98-105.
- Read Michael Crowe, "The Equant," p. 38.
- Watch this short video on Geometrical models.
- Optional: photos of Ptolemy's Almagest, first printed edition by Regiomontanus (1496).
- Explore these animated models of various geometrical devices to ensure that
you understand them: | Equant
system for outer planet on epicycle; Earth eccentric; with Sun | Craig
Sean McConnell (scroll down to equant animation) | Dennis
Duke, click on Ptolemy's Moon |
- Read Crowe, pp. 42-49, the introductory section of his chapter on Ptolemy's
astronomy. Don't worry about all the geometrical details; read for the gist
of Crowe's points about Ptolemy. These are the true-false study questions
you will answer in this section:
- Ptolemy, author of the greatest book of ancient mathematical astronomy,
was a king of Egypt.
- Ptolemy wrote the Almagest in Arabic.
- What Euclid's Elements was to geometry, Ptolemy's Almagest
was to astronomy.
- Ptolemy calculated the distances to the planets in the Almagest.
- In addition to the Almagest, Ptolemy wrote mathematical treatises on
optics, music theory, and geography.
- Ptolemy wrote a manual of astrology called the Tetrabiblos.
- In Ptolemy's astronomical system, every planet was integrated into one
single model, rather than treated independently.
- Ptolemy employed eccentric circles and epicycles in his astronomical
models.
- Ptolemy's lunar models accurately predicted the position of the Moon
but not its apparent diameter (angular width).
- In an equant model, motion is uniform with respect to the center of
a circle.
- In an equant model, motion is uniform with respect to an observer on
the eccentric Earth.
- In Ptolemy's models, the motion of the outer planets was related to
the motion of the Sun, because the radius of each planetary epicycle was
set parallel to the line from the Earth to the Sun.
- In Ptolemy's models, the motion of the inner planets (Venus and Mercury)
was related to the motion of the Sun, because the center of each planetary
epicycle was set on the same line (collinear) as the line from the Earth
to the Sun.
- Read the excerpts from the Preface and Book I of Ptolemy's Almagest
found in Crowe, pp. 50-65. These are the true-false study questions you will
answer in this section:
- Ptolemy believed that the mathematical astronomer teaches beautiful
theories.
- Ptolemy classified theoretical sciences into theology, physics, and
mathematics, where mathematics falls in between the other two.
- Ptolemy argued that astronomy, physics and theology each could attain
certain knowledge in their own subject areas.
- Ptolemy argued that astronomy could help both theology and physics in
achieving knowledge.
- Ptolemy argued that astronomy inculcates a love of the eternal and unchanging,
and thereby a love of divine beauty.
- Ptolemy argued that the fixed stars that daily rise and set move like
a rotating sphere.
- Ptolemy argued that the rising and setting of fixed stars provides evidence
that the Earth is spherical.
- Ptolemy argued that the Earth lies at the center of the universe because
the celestial equator, the plane of the horizon, and other great circles
bisect the heavens into equal hemispheres.
- Ptolemy argued that the universe is relatively small compared to the
size of the Earth.
- Ptolemy argued that earth falls downward from all sides toward the center
of the universe.
- Would you have used the equant, or not?
The equant was a very powerful device, by which Ptolemy was able to achieve
an unparalleled degree of accuracy in quantitative predictions
of the positions of the planets. However, this accuracy came at a price, because
there was a notable problem with the equant that bothered many astronomers.
In a rotating sphere, the axis of rotation runs through the center of the
sphere. Yet Ptolemy’s equant requires a sphere to rotate uniformly around
a point not on its axis, which is mechanically impossible
for a rigid sphere. Review the definitions of an equant model provided by
Lindberg and Crowe; do you see why this is the case? What ramifications might
this have held for the concept of solid celestial spheres?
- Further explanation of this difficulty: It is impossible to model an equant with a styrofoam ball. A sphere is defined by a point
and a given radius that sweeps out a circumference in 3 dimensions. A
rotating sphere has an additional entity, an axis, that passes through
the center. If one changes any of those three entities (center, radius,
axis of uniform rotation) one no longer has the same sphere. Yet the equant
model does precisely this: it calls for a sphere to rotate uniformly around
a point not on its axis (the equant). One might as well try to shift the
center to the equant, shorten or lengthen the radius on the fly, and smoosh
the circumference as the sphere continues to rotate. The only way to do
it is to make the sphere elastic or fluid, like a nerf ball being pushed around from one side to another within a steel spherical shell;
it’s mechanically impossible for a rigid physical sphere to move
this way. And if it does move this way, by definition it's not really
the same sphere anymore, because the center, radius, and axis are not constant.
- Noel Swerdlow and Otto Neugebauer, Mathematical Astronomy in Copernicus’s
De Revolutionibus (Springer-Verlag, 2 vols) explain this in a summary
point about Ptolemy's astronomy (p. 41):
Ptolemy’s “Planetary Hypotheses contain physical
representations of the models that are supposed to exist in the heavens
and produce the apparent motions of the planets, but these lead to difficulties,
the most notable being the violation of uniform circular motion by spheres that are required to rotate uniformly with respect to points
not located on their axes, and thus rotate nonuniformly about their axes.”
Further on (pp. 43-44), they summarize the Arabic critique of the equant:
“...when Ptolemy's planetary model is physically represented as
complete spheres, a problem arises in that the sphere carrying the epicycle
must rotate uniformly around a point, the equant point, not on its axis,
so that the motion of the sphere with respect to its axis is nonuniform.
The serious physical, or mechanical, problem is that there is no way of
compelling a sphere, a ‘simple body’ in the Aristotelian sense,
to do this all by itself.”
"I
know that I am mortal and living but a day.
But when I search for the numerous turning spirals of the stars,
I no longer have my feet on the Earth,
But am beside Zeus himself,
filling myself with god-nurturing ambrosia."
Anonymous epigraph often attributed to Ptolemy