1 The Birth of Science - MSRI

56 3. Other Hellenistic Scientific Theories
attested by Apuleius 32 and by Theon. 33 It is reasonable to think that in
this book Archimedes, who had written theoretical works on parabolas
and paraboloids and had even, as we shall see, applied hydrostatics to
paraboloids, would mention the caustic properties of parabolically shaped
mirrors — in fact Apuleius, listing some of the book’s contents, explicitly
mentions concave mirrors able to concentrate the sun’s rays on one
point. 34 One can see how, combining such writings of Archimedes with
the recollection of his contribution to the defense of Syracuse, which included
the construction of ballistic weapons capable of setting fire to ships
from far away, the traditional belief may have arisen.
The most interesting theoretical result about reflection is probably a theorem
contained in Heron’s Catoptrics, 35 according to which a light ray that
leaves a point A and reaches a point B after reflection in a plane mirror
follows the shortest path from A to B that touches the mirror. 36 The reflection
law can thus be deduced from a minimization principle — the oldest
such principle known. Note also that Archimedes had already deduced
the laws of reflection from the principle of reversibility of optical paths. 37
Ptolemy’s Optics is the earliest extant work that includes a systematic page 89
account of refraction phenomena. 38 But studies of refraction started much
earlier; even Ptolemy’s observation that a heavenly body is seen elsewhere
than in the true direction where it lies because of atmospheric refraction
seems to go back to Hellenistic times. 39
Ptolemy’s Optics also tabulates the refraction angles corresponding to
various incidence angles for air-water, air-glass, and water-glass interfaces.
40 Apparently Ptolemy thought that the refraction angle varies with
the incidence angle according to what we call a quadratic function. He
does not state the functional dependence explicitly, but the values he gives
32
Apuleius, Apologia, xvi.
33
Theon, Commentary on Book I of the Almagest, [Theon/Rome], pp. 347–349.
34
Apuleius, loc. cit.
35
De speculis, 4. This work, preserved anonymously in Latin and reproduced in [Heron: OO],
vol. II, part i, is believed to be a translation of Heron’s Catoptrics.
36
The proof of this is very simple: it is based on the observation that the path followed by the
ray does not change in length if the first portion, going from A to the incidence point, is replaced
by its mirror image.
37
Archimedes’ proof is reported in a scholium to the pseudo-Euclidean Catoptrics ([Euclid: OO],
vol. VII, p. 348, sch. 7).
38
All we have of this work is an incomplete and often obscure Latin translation made
in the twelfth century from an Arabic version. The Latin translation was first published in
[Ptolemy/Govi]; the critical edition is [Ptolemy/Lejeune].
39
Ptolemy, Optics, V, 23–30 and 237, 20 – 242, 7 (ed. Lejeune); the same observation appears in
Cleomedes, Caelestia, II, 6, 174–177 (ed. Todd). Compare also Sextus Empiricus, Adversus mathematicos,
V, 82. Other mentions of refraction appear in works from the early imperial period, such
as Seneca, Naturales quaestiones, I, vi, 5.
40
Ptolemy, Optics, V, 7–21 = 227, 1 – 237, 7 (ed. Lejeune).
Revision: 1.13 Date: 2002/10/16 19:04:00