Full Idea
Euclid gives no principle of continuity, which would sanction an inference that if a line goes from the outside of a circle to the inside of circle, then it must intersect the circle at some point.
Gist of Idea
Euclid needs a principle of continuity, saying some lines must intersect
Source
comment on Euclid (Elements of Geometry [c.290 BCE]) by Stewart Shapiro - Philosophy of Mathematics 6.1 n2
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.183
A Reaction
Cantor and Dedekind began to contemplate discontinuous lines.