Dectrine of Parallels. 91 
With respect to the analytical proof proposed by the same au- 
thor, based upon the theory of functions and the principle of ho- 
mogeneity, I should have called it hardihood to defend its 
conclusiveness against the undeniable objection of Prof. Lesuir, 
that the identical argument by which Lecenpre attempts to show 
that the third angle of a triangle is determined by the other two, 
would show that the third side is also determined alone by the 
other two, had not the attempt to defend its conclusiveness been 
made both by its author and by men in the same rank of talent. 
To what has been written on that subject, 1 may be permitted, 
perhaps, to add one or two brief inquiries, as follows: First, how 
conclusive so ever the proof in question may appear to analysts of 
a certain practiced and subtle penetration, yet, inasmuch as the 
purpose of a demonstration is to manifest truth to those who are 
in the capacity of being taught, can it be demanded of ordinary 
reasoners to conclude that, because two triangles having two equal 
angles adjacent to one equal side would coincide, and have their 
third angles equal, that, therefore, if the side were varied, the oth- 
er two sides would meet as before ; and, moreover, that the third 
angle in this and the former triangles would be one and the same 
function of the side and the angles adjacent to it,—that is to say, 
that there is some specific arithmetical process by which the third 
angle may be deduced from the three quantities or elements na- 
med? All this, however, is demanded in the outset of the proof 
under consideration. Again: it will be conceded, I presume, that 
no property peculiar to straight lines can be demonstrated with- 
out the introduction somewhere in the proof, either explicitly or 
implicitly, of at least one step dependent on some property known, 
either by definition or by antecedent evidence, to pertain exclu- 
sively to such lines. My second question, therefore, would be, 
what one step of the so called analytical argument embodies any 
such peculiar property ? 
_ Among Lecenpre’s methods of proof, there is one proposed in 
a note to Prop. x1x, of the 12th edition of his Geometry, which 
depends upon the equal division of an infinite plane by an infinite 
straight line, and the consequent necessity inferred in the argu- 
ment, that a straight line contained in an angle must, if produced, 
eventually meet the containing sides (one side it should be) of the 
angle. But the argument, as Mr. 'T. P. Taompson in his exam- 
ination has pointed out, equally proves that two parallels can not 
