Barth’s use of the tangent metaphor is very powerful, but assumes a reader who remembers Euclidean plane geometry. That subject is now taught differently than it probably was in Swiss academic gymnasia in the late 19th century. Euclid wrote about what came to be called the tangent, Elem. III def. 2-3:
Εὐθεῖα κύκλου ἐφάπτεσθαι λέγεται, ἥτις ἁπτομένη τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνει τὸν κύκλον. Κύκλοι ἐφάπτεσθαι ἀλλήλων λέγονται οἵτινες ἁπτόμενοι ἀλλήλων οὐ τέμνουσιν ἀλλήλους.
Euclidis Elementa, ed. Heiberg, Teubner 1883; transl. Heath, 2nd. ed.
[A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. Circles are said to touch one another which, meeting one another, do not cut one another.]
If this seems less than obvious, keep in mind Euclid’s definite of a point, Elem. I. def. 1:
Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
[A point is that which has no part.]
[Citations as above]
Thus a tangent “touches” a circle only at one point, but since the point has no part, it is infinitesimal –hence the logical paradox: the line touches the circle only at one infinitesimal point, and hence does not touch the circle. Another explanation by D.E. Joyce of Clark University can be found here. An interactive demonstration of a tangent can be found Math Open Reference(N.B. uses Java).
Barth’s use of the tangent metaphor is slightly more literary than strictly accurate –the tangent does touch the circle but only at a point which has no “part” –that is, which is an infinite indivisible, and by extension, justification for Barth’s “touch / does not touch” language.