A history of Greek mathematics - Wilbourhall.org

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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.