OPTICS 
443 
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appears to be a circle; if the eye approaches nearer to 
the sphere the portion seen becomes less, though it appears 
greater ; if we see the sphere with two eyes, we see a hemi 
sphere, or more than a hemisphere, or less than a hemisphere 
according as the distance between the eyes is equal to, greater 
than, or less than the diameter of the sphere; these pro 
positions are comparable with Aristarchus’s Proposition 2 
stating that, if a sphere be illuminated by a larger sphere, 
the illuminated portion of the former will be greater 
than a hemisphere. Similar propositions with regard to the 
cylinder and cone follow (Props. 28-33). Next Euclid con 
siders the conditions for the apparent equality of different 
diameters of a circle as. seen from an eye occupying various 
positions outside the plane of the circle (Props. 34-7) ; he 
shows that all diameters will appear equal, or the circle will 
really look like a circle, if the line joining the eye to the 
centre is perpendicular to the plane of the circle, or, not being 
perpendicular to that plane, is equal to the length of the 
radius, but this will not otherwise be the case (35), so that (36) 
a chariot wheel will sometimes appear circular, sometimes 
awry, according to the position of the eye. Propositions 
37 and 38 prove, the one that there is a locus such that, if the 
eye remains at oiqe point of it, while a straight line moves so 
that its extremities always lie on it, the line will always 
capear of the same length in whatever position it is placed 
(not being one in which either of the extremities coincides 
with, or the extremities are on opposite sides of, the point 
at which the eye is placed), the locus being, of course, a circle 
in which the straight line is placed as a chord, when it 
necessarily subtends the same angle at the circumference or at 
the centre, and therefore at the eye, if placed at a point of the 
circumference or at the centre ; the other proves the same thing 
for the case where the line is fixed with its extremities on the 
locus, while the eye moves upon it. The same idea underlies 
several other propositions, e.g. Proposition 45, which proves 
that a common point can be found from which unequal 
magnitudes will appear equal. The unequal magnitudes arc 
straight lines EG, CD so placed that BCD is a straight line. 
A segment greater than a semicircle is described on EG, and 
a similar segment on CD. The segments will then intersect