The axioms of descriptive geometry, by A.N. Whitehead.

40 THE CORRESPONDENCE OF NEIGHBOURHOODS [CH. IV
38. If a reentrant single-branched curve a (which may be a straight
line) is transformed by an infinitesimal transformation of a continuous
group into a curve /f, then the senses* round, or directions round, the
curves correspond in a perfectly definite manner, the same for all such
infinitesimal transformations.
In order to make clear the correspondence of directions round any
two reentrant single-branched curves a and /f, let OP1L and OP2L
define two complementary segments round a, and let O'Q1M and O'Q2M
define two complementary segments round P. Now consider any
one-one point transformation which (1) transforms a into /f, (2) transforms segments of a into segments of fi, (3) transforms O into 0'.
Then one of the two following mutually exclusive cases must hold, either
one of the two, O'QM and the relatum of O.PL, contains the other, or
one of the two, O'Q2M and the relatum of OP1L contains the other.
If one of the two, O'QjM and the relatum of OP1L, contains the other,
then the segments OP1L and O'QM will be said to correspond in sense
where O and 0' are corresponding origins. Also we shall consider an
arbitrary small portion of a containing O as the neighbourhood of O;
thus O divides its neighbourhood into two parts, one lying in the
segment OP1L, and the other in the segment OP2L. Similarly O'
divides its neighbourhood on f8 into two parts. Then the case contemplated above, when the segments OP1L and O'Q1M correspond in
sense with O and 0' as corresponding origins, will also be expressed by
saying that O corresponds to 0' and the neighbourhood of O in the
segment OPL corresponds to the neighbourhood of O' in the segment
O'Q1MI.
Now considering the case of an infinitesimal transformation, the
curve /f must lie infinitesimally near to the curve a, so that the point
Q1 may be assumed to be a point infinitesimally near to the point Pi
and the point Q2 to be a point infinitesimally near to the point P2.
Then no point of the segment OP1L which is infinitesimally near to P1
is infinitesimally near to any point on the segment O'Q2M. Hence the
segments OP1L and O'Q[M must correspond in sense with O and 0' as
corresponding origins. Thus only one of the two cases of correspondence in sense is now possible.
Notice that for this theorem the curves a and f need not be distinct,
nor need the points O and 0'.
If a straight line I is latent for a transformation, and O is a latent
point on it, and segments with origin O correspond in sense with
* Cf. Proj. Geom. ~ 15, extended to any reentrant lines.