94 Dectrine of Parallels. 
which has baffled the skill applied to it in so many forms, can le- 
gitimately spring, I would propose the inquiry whether, in case 
some one of the more recondite truths of geometry—as, for example, 
the ratio of a sphere to its circumscribing cylinder—should, by 
a flash of reasoning and through a brief step or two, be connected 
with the ordinary and unexceptionable definitions, the latter might 
not, just as unavoidably, seem inadequate to contain or give ori- 
gin to such a conclusion as the former ? 
Proposition. 
Two straight lines which make, with a third line that cuts them, 
the two interior angles together less than two right angles will 
meet, if indefinitely produced. 
Demonstration.—Let the straight Fig. 1. 
line AD (fig. 1) divide the angle yi 
C; and, if it divides it une- 
qually, cut off from the greater 
part, as DAB, an equal, DAE, to 
DAC the less. 
' ‘The straight line AD either be- 
longs to, or, in other words, is con- 
tained by the angular space DAC 
or it is mo¢ contained by it. And 
if, by the nature of an angular 
space, it 7s contained by DAC it must evidently be, in like man- 
ner, contained by DAE; and if not contained by DAC it is not con- 
tamed by DAE; but, if contained by DAE, the construction 
of an adjoining ongis EAB cannot affect that fact nor, mutatis 
mutandis, the contrary. We therefore see that, if a line divides 
an angle, it must be contained by both parts or by neither,—so 
that either the two coincide in that line or are separated by it. 
On whatever condition, therefore, an angle, as DAC, shall have 
been constituted, (as, 83 example, by drawing AD and AC through 
ed points, ) it is not allowable to constitute an adjoining angle 
DAB on such a condition as shall exclude AD from either being 
contained by both angular spaces or else, by neither. 
Let then EG (fig. 2) be an indefinite straight line, and Aa 
bint without it. Let AB be a line cutting EG at right angles in , 
an l produce BA to C. 
= 
a 
