A steep climb up a mountain is sometimes described as “vertiginous”, as in the following quotation from a description of a hike up the Inca Trail:
You have time to make the vertiginous climb to its summit for dramatic views of the city spread out below.
You might be excused for thinking that “vertiginous” is related to “vertical”, perhaps with the additional implication of being precarious and dangerous. In fact, the two words are from different roots and have quite distinct meanings.
Vertical, which is the adjective form of the English noun vertex, comes originally from the identical Latin word vertex, meaning an eddy or a summit. A line is vertical if it rises to a vertex, perpendicular to the horizon. So, a vertical cliff is one that goes straight up from the ground.
Vertiginous, meanwhile, means dizzying; it’s the adjectival form of the noun vertigo, meaning dizziness or giddiness. It derives from the Latin word vertigo, meaning whirling. So, an ascent would not have to be vertical in order to be vertiginous and, if you weren’t prone to vertigo, it could be vertical and not vertiginous.
Some dictionaries suggest that the Latin words vertex and vertigo do share a common root : vertere, meaning to turn. It’s easy to see how vertiginous has evolved from this meaning, with its sense of dizzy whirling. Vertical, meanwhile, presumably derives because something could rotate around a vertical axis.