Doctrine of Parallels. 95 
Fig. 2. 
c 
1B 
Let the angle BAD be constituted by the condition that it can 
contain all the lines drawn through A that can meet FG infinitely 
produced on the one side of CBtowardsG. Then I say that the 
angle DAC cannot be constituted simply on the condition that 
it can contain all the lines that will not meet FG produced as 
before; for, then, whether AD be contained by both BAD and 
DAC, or by neither, it must meet and not meet FG at the same 
time; which is absurd. In searching, therefore, for the proper 
constitution of the angle DAC, we observe that, if some point, 
as E, on the other side of BC from G be united with A, and the 
line be produced, that line will lie in the angular space DAC ; also, — 
if a line be drawn at right angles to CB through A, it will not 
meet on either side; for if it be supposed to meet on one side, 
then, for the same reason, it must meet on the other; which is 
impossible. _ 
There subsists, therefore, a threefold distinction in the lines that 
can be drawn in the entire angular space on the one side of BC 
towards G; first, lines that can meet the line EG on that one 
side ; second, lines that can meet the same on the other side ; and 
third, a line or lines that can meet on neither side ; also, it is evi- 
dent, that in relation to meeting EG, these are all the distinctions 
that can subsist. 
If, now, there can subsist more lines than one through A that 
meet on neither side, let the angle CAH be constituted so as to 
contain all that can meet on the other side from G, and of course, 
HAD must contain all that can meet on neither side. Now if 
AD is contained by BAD, it is aso contained by DAH,—that is, _ 
it meets on the one side, towards G, and on-neither side at the same 
