Reply to the criticism on Demonstration of Parallels. 71 
not weaken the argument; for since the angle BAD is expressly 
conditioned to contain all the lines of a certain class, and only 
such lines, it follows that if it cannot be predicated of AD that it 
is contained in the angle it is not one of the class, and vice 
versa. 
Turning now to the other part of the objection, in the second 
member of the sentence above quoted, beginning with the phrase 
“in other words,” I maintain it to embody not other words 
merely, but other ideas. Not only the two clauses convey prop- 
ositions that are not identical, but the latter is not deducible from 
the former, neither does it seem relevant to an argument which 
turns not upon ideas of “ limits” or limiting lines, under a classifi- 
cation simply logical, but upon a certain natural or geometrical 
relation of a line dividing an angular space to the component 
parts of the space it divides. 
If, therefore, no more weighty objection than these or the 
“similar” ones adverted to can be adduced by a critic of such un- 
questionable discernment as the one whose authorship I perceive 
in his initials, it may be counted asa new symptom, added to 
those remarked upon in the original paper, of the genuineness of 
the reasoning ; and with respect to the epithet “plausible,” ap- 
plied by the Editors of this Journal—in a manner entirely cour- 
teous I acknowledge—to that reasoning, I should not hesitate to 
transfer it, in a manner not wncourteous I trust, to the objections 
urged against it. 
If there is a fallacy in the attempted demonstration, it is to be 
found, as I judge, after carefully revising the method, in the fol- 
lowing conclusion near the bottom of p. 95, “and, of course, 
HAD must contain all that can meet on neither side.” If an 
objector should urge that, although under the supposition of more 
lines than one that meet on neither side, HAD might be consti- 
tuted, independently, to contain them a// and no others, yet that 
such a constitution could not co-exist or be compatible with the 
antecedent constitution or definition of BAD and DAC, I ac- 
knowledge that the best answer I could give, at present, does not 
seem entirely satisfactory. 
‘May 5th, 1846. 
