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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 Y 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) corre 
sponding to different angles of incidence (i); these are calcu 
lated on the supposition that r and i are connected by an 
equation of the following form, 
r =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 O r ptics in the Arabic at once 
made it clear that the work De speculis 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 Auto 
mata and jDioptra, and partly by a quotation by Damianus 
{On hypotheses in Optics, chap. 14) of a proposition proved by 
‘ the mechanician Heron in his own Cptoptrica ’, 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 1 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.