A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

in
THE PLAN1SPEAERIUM OF PTOLEMY 293
of oblique circular cones has led to the conjecture that Apollonius
was the discoverer of the method. But Ptolemy makes no
mention of Apollonius, and all that we know is* that Synesius
of Gyrene (a pupil of Hypatia, and born about a.d. 365-370)
attributes the discovery of the method and its application to
Hipparchus ; it is curious that he does not mention Ptolemy's
treatise on the subject, but speaks of himself alone as having
perfected the theory. While Ptolemy is fully aware that
circles on the sphere become circles in the projection, he says
nothing about the other characteristic of this method of projection,
namely that the angles on the sphere are represented
by equal angles on the projection.
We must content ourselves with the shortest allusion to
other works of Ptolemy. There are, in the first place, other
minor astronomical works as follows
(1) $d(T€is anXavtov da-repcou of which only Book II survives,
(2) 'TTTodiareis t&v TrXaucofxeucou in two Books, the first
of which is extant in Greek, the second in Arabic only, (3)
inscription in Canobus, (4) Ilpoxeipcoy kclvovcdv SiaTacns kcu
yjrr)(po(popia. All these are included in Heiberg's edition,
vol. ii.
The Optics.
Ptolemy wrote an Optics in five Books, which was translated
from an Arabic version into Latin ' the twelfth
century by a certain Admiral Eugenius Siculus * ; Book
the
I,
however, and the end of Book V are wanting. Books I, II
were physical, and dealt with generalities ; in Book III
Ptolemy takes up the theory of mirrors, Book IV deals with
concave and composite mirrors, and Book V with refraction.
The theoretical portion would suggest that the author was
not very proficient in geometry. Many questions are solved
incorrectly owing to the assumption of
a principle which is
clearly false, namely that the image of a point on a mirror '
is
at the point of concurrence of two lines, one of which is drawn
from the luminous point to the centre of curvature of the
mirror, while the other is
the line from the eye to the point
See G. Govi, L'ottica di Claudio Tolomeo di Euyenio Ammiraglio dA
1
Skilia, ... Torino, 1884; and particulars in G. Loria. Le scienze e*atte
nelV antica Grecia, pp. 570, 571.
294 TRIGONOMETRY
on the mirror where the reflection takes place
'
; Ptolemy uses
the principle to solve various special cases of the following
problem (depending in general on a biquadratic equation and
now known as the problem of Alhazen), ' Given a reflecting
surface, the position of a luminous point, and the position
of a point through which the reflected ray is required to pass,
to find the point on the mirror where the reflection will take
place.' Book V is the most .interesting, because it seems to
be the first attempt at a theory of refraction. It contains
many details of experiments with different media, air, glass,
and water, and gives tables of angles of refraction (r) corresponding
to different angles of incidence (i) ; these are calculated
on the supposition that r and i are connected by an
equation of the following form,
r = ai — bi
2
,
where a, b are constants, which is worth noting as the first
recorded attempt to state a law of refraction.
The discovery of Ptolemy's Optics in the Arabic at once
made it clear that the work Be specvlis formerly attributed
to Ptolemy is not his, and it
is now practically certain that it
is, at least in substance, by Heron. This is established partly
by internal evidence, e.g. the style and certain expressions
recalling others which are found in the same author's Automata
and Dioptra, and partly by a quotation by Damianus
(On hypotheses in Optics, chap. 14) of a proposition proved by
'
the mechanician Heron in his own Catoptrica ', which appears
in the work in question, but is not found in
Ptolemy's Optics,
or in Euclid's. The proposition in question is to the effect
that of all broken straight lines from the eye to the mirror
and from that again to the object, that particular broken line
is shortest in which the two parts make equal angles with the
surface of the mirror; the inference is that, as nature does
nothing in vain, we must assume that, in reflection from a
mirror, the ray takes the shortest course, i.e. the angles of
incidence and reflection are equal. Except for the notice in
Damianus and a fragment in Olympiodorus l containing the
proof of the proposition, nothing remains of the Greek text
1
Olympiodorus on Aristotle, Meteor, iii. 2, ed. Ideler, ii, p. 96, ed.
Stiive, pp. 212. 5-213. 20.