Ptolemy appropriated from his predecessors is difficult to determine because of Ptolemy’s
silences and the dearth of other Greek technical literature on the subject, and there is
no modern consensus.
A byproduct of Ptolemy’s determining celestial models in the Almagest is a set of tables,
interspersed among the chapters of the Almagest, enabling computation of a full range of
phenomena including the positions of heavenly bodies at any given date. Subsequently,
Ptolemy published a revised and expanded set of tables as the Handy Tables, used extensively
in antiquity and the Middle Ages, especially by astrologers. Ptolemy gave a more physical
account of the models in Planetary Hypotheses, in two books, surviving complete only in
Arabic translation. He returned here to a question that he had regarded as inconclusive in
the Almagest, the distances of the planets, proposing a system of nested and contiguous
systems of etherial spheres in the order (outward from the Earth): Moon, Mercury, Venus,
Sun, Mars, Jupiter, Saturn, stars. By way of Islamic astronomy this became the standard
cosmological model until the 16th c.
Ptolemy’s other astronomical writings are relatively minor. Phaseis is a parape ̄gma,
arranged according to the solar year and the principal latitudes of the Greco-Roman world.
Planispherium (extant only through Arabic translation) is a study of stereographic projection,
a mathematical technique for representing circles on the celestial sphere by circles in a
plane, the basis of the primary astronomical instrument of late antiquity and the Middle
Ages, the plane astrolabe. Analemma, extant in fragments in Greek and a more or less com-
plete Medieval Latin translation, concerns the mathematical theory underlying sundials.
Ptolemy also appears to have written monographs, now lost, on the theory of visibility of
stars and of the planets Venus and Mercury.
Other sciences: The Harmonics, in three books, deduces models for systems of tuning
employed by Greek musicians. Probably one of his earliest treatises, it contains discussions
of scientific epistemology that have bearing on Ptolemy’s work in other sciences. Ptolemy
situates his own harmonic theory in relation to two faulty theories: that of the Pythagor-
eans, which modeled the intervals in Greek scales by means of a highly restricted set of
ratios of whole numbers, and that of the Aristoxeneans, which was ostensibly empirical and
eschewed ratios. These theories complementarily exemplify reason insufficiently controlled
by the senses, and empiricism insufficiently controlled by reason, though Ptolemy’s solution
falls closer to the Pythagoreans by embracing a more flexible system of whole-number
ratios. Ptolemy’s central claim is that the general constraints he proposes for the ratios of
musical intervals within scales lead to a finite set of possible scales that is almost coextensive
with the scales employed by contemporary musicians. Experimental apparatus plays an
important role in this complex work.
The Optics, in five books, is a study of the phenomena of visual perception, including long
treatments of binocular vision and the appearances of objects seen reflected in mirrors or
refracted through the interfaces between different transparent media. It unfortunately sur-
vives only in a defective Medieval Latin translation of a lost Arabic translation, lacking
the whole of Book 1 and end of Book 5. Ptolemy’s model for vision assumes that it is
effected through a cone-shaped visual ray with its vertex at the eye; perception occurs along
straight lines emanating from the cone’s vertex. Ptolemy probably thought of the ray as an
alteration of the exterior environment caused by the human soul. In contrast to the
model of visual rays in E’s Optics, there are no gaps between potential lines of
sight in Ptolemy’s cone, and it has some ability to perceive the distance of objects. Again
Ptolemy makes appeal to experimental apparatus, most impressively on binocular vision
PTOLEMY