96 Doctrine of Parallels. 
time; which is absurd. But if AD is not contained by BAD it 
does not meet on the one side towards G,—yet it cannot (by 
what was proved at the first) be contained by DAH, and there- 
fore must meet on the one side or the other, and therefore, since 
that is not on the one side towards G, it must be on the other to- 
wards E:: therefore, AD is contained by the angular space HAC, 
from which it is separated by the angular space HAD; which 
is impossible. Therefore there cannot be any such angular space 
HAD; but AH and AD must coincide, and there can be but one 
line through A, that does not meet EG on either side, and that, as 
has been shown, is the line at right angles to CB. 
Now it is well known that if a line cut two others so as to make 
the two interior angles together equal to two right angles, and if 
the cutting line be bisected, and from the bisecting point a perpen- 
dicular to one of the two lines be drawn, it may be and is proved 
perpendicular to the other. But it cannot be perpendicular to 
any other line through either extremity of the cutting line—else 
one and the same triangle might have two angles both right an- 
gles, which is impossible. Therefore, any other lines than those 
which make, with a third, the two interior angles together ae: 
to two right angles will meet, if indefinitely produced. 
Corotiary.— The sides of triangles are not contained by the 
triangular spaces, but are mere dividing lines between the space 
within and the space without. Also the same is true of all super- 
ficial figures. Also the surfaces of solids are not contained by the 
solids, but are dividing surfaces between the space within and the 
space without. 
Fig. 3. For let ABC (fig. 3) be a tn- 
angle. Produce BA to D, Now 
if the only actual distinction that 
subsists among all the lines pass 
ing through A, with reference to 
the limited line BO, is that lines 
of one class meet or intersect it and 
c those of another do not, we have, 
on ae one side of BD towards C, the angle BAC constituted 
or defined as that which can contain the first only, and the angle 
CAD as that which can contain the second only. Whether, 
ther fore, the line AC is contaisfed in both angular spaces or con- 
din neither, the absurdity arises that it must intersect and 
