442 
EUCLID 
magnitudes ’ ; Def. 2 : ‘ The hgure contained by the visual rays 
is a cone which has its vertex in the eye, and its base at the 
extremities of the objects seen’; Def. 3: ‘And those things 
are seen on which the visual rays impinge, while those are 
not seen on which they do not ’ ; Def. 4 : ‘ Things seen under 
a greater angle appear greater, and those under a lesser angle 
less, while things seen under equal angles appear equal ’ ; 
Def. 7 : ‘ Tilings seen under more angles appear more distinctly.’ 
Euclid assumed that the visual rays are not 1 continuous ’, 
i.e. not absolutely close together, but are separated by a 
certain distance, and hence he concluded, in Proposition 1, 
that we can never really see the whole of any object, though 
we seem to do so. Apart, however, from such inferences as 
these from false hypotheses, there is much in the treatise that 
is sound. Euclid has the essential truth that the rays are 
straight ; and it makes no difference geometrically whether 
they proceed from the eye or the object. Then, after pro 
positions explaining the differences in the apparent size of an 
object according to its position relatively to the eye, he proves 
that the apparent sizes of two equal and parallel objects are 
not proportional to their distances from the eye (Prop. 8) ; in 
this proposition lie proves the equivalent of the fact that, if cx, 
are two angles and ex < /3 < tt, then 
tan cx cx 
tan /3 < /3 ’ 
the equivalent of which, as well as of the corresponding 
formula with sines, is assumed without proof by Aristarchus 
a little later. From Proposition 6 can easily be deduced the 
fundamental proposition in perspective that parallel lines 
(regarded as equidistant throughout) appear to meet. There 
are four simple propositions in heights and distances, e.g. to 
find the height of an object (1) when the sun is shining 
(Prop. 18), (2) when it is not (Prop. 19) : similar triangles are, 
of course, used and the horizontal mirror appears in the second 
case in the orthodox manner, with the assumption that the 
angles of incidence and reflection of a ray are equal, ‘as 
is explained in the Catoptrica (or theory of mirrors) ’. Pro 
positions 23-7 prove that, if an eye sees a sphere, it sees 
less than half of the sphere, and the contour of what is seen