93 Doctrine of Parailels. 
tain certain lines, be defined as that which shall contain or be 
full of some physical substance, gold for example, while the ad- 
joining angular space DAB, be defined to be full of some other, 
as silver, it is plain at once that the line AD is simply a dividing 
line between the gold and the silver. And—demonstration aside 
—reflection will perhaps make it apparent that an abrupt transi- 
tion from that which can contain the lines of one specific proper- 
ty to that which can contain the lines of an opposite or distinct 
property, can no more take place, except through a peculiar or di- 
viding line, than from that which is full of gold to that which is 
full of silver. 
It is essential to observe farther, a twofold, but obvious requi- 
site as to the distinctions that can be employed. First, they must 
not be arbitrary—that is to say, such as either have no pertinence 
to the point at issue, or do not define the spaces : Second, a specific 
distinction being once established as a basis of the argument, 70 
correlative or homogeneous distinction that can subsist must be 
overlooked. 'Thus, in the corollary, if BC be produced and an 
angle adjoining BAC be constituted as that which can contain 
all the lines that will intersect beyond C, reason shows two correl- 
atives, neither of which may be neglected, so that the three will 
stand, beyond C, at C, and on this side of C. 
Finally, in the application of our elementary principle or re 
lation to analytical geometry, the correlative distinctions of lines 
in a given case will be perceived to be threefold, manifold or 
even unlimited, according to the conditions of the application. 
In the case of lines through a point without a circle they would 
be threefold—lines that eut the circumference, lines that touch 
and lines that are capable of neither—or, otherwise they might 
be, lines that. cut in fo points, in one, and in no point. In the 
ease of curves with different branches they might be manifold, 
and in that of a spiral cut by an unlimited straight line they 
miust of necessity be unlimited. Whether, in any of the possible 
applications, valuable truths, other than the two I have developed, 
would be the result, there has not yet been opportunity sufficiently 
to consider. — 
Middlebury, June, 1845. 
_ [For an editorial note to Prof. Twisixa’s article, see p. 147 of this No.] 
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