Parans
Parans at your latitude
A paran is two planets angular at the same moment: one rising as another culminates, and so on. Unlike a line it belongs to a latitude rather than a place, so it holds right around the world at that latitude. It is also the part of the technique that has something to say when no line runs anywhere near you.
Why this works where a map does not
Most of the Earth is nowhere near any of your forty lines, which makes the commonest honest answer an astrocartography map can give about a specific place nothing in particular. Parans are the exception: they apply to a whole band of latitude, so somewhere with no line through it can still have several.