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THIED CLASS OF OBJECTS FOE THE SUBJECT. 163
of tlie other, each to each, to which the equal sides are
opposite ; therefore the angle I a e is equal to the angle
e c f.
But the angle e c d is greater than the angle e c /
Therefore the angle a c d is greater than the angle a b c"
" In the same manner, if the side b c be bisected, and the
side a c be produced to g, it may be demonstrated that the
angle beg, that is, the opposite vertical angle a c d is
greater than the angle ab e."
My demonstration of the same proposition would be as
follows (see fig. 5) :
For the angle b a c to be even equal to, let alone greater
than, the angle a c d, the line b a toward c a would have to
lie in the same direction as b d (for this is precisely what
is meant by equality of the angles), i.e., it must be parallel
Fig. 5.
with b d ; that is to say, b a and b d must never meet ; but
in order to form a triangle they must meet (reason of
being), and must thus do the contrary of that which would
be required for the angle b a c to be of the same size as
the angle a c d.
For the angle a b c to be even equal to, let alone greater
than, the angle a c d, line b a must lie in the same direction
towards b d as a c (for this is what is meant by equality of
the angles), i.e., it must be parallel with a c, that is to say,
b a and a c must never meet ; but in order to form a triangle
b a and a c must meet and must thus do the contrary of
that which would be required for the angle a b c to be
of the same size as a c d.
By all this I do not mean to suggest the introduction of