All 4 Uses

vengeance
in
The Magic Mountain
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Ottilie came down with a high fever, let her chart just shoot up with a vengeance.†
Chpt 3.9 *

with a vengeance = with intensity
Sneezing was just as much fun—the way you felt it swelling up with a vengeance inside you, until it became irresistible and you breathed in and out in one great frenzy, gave yourself over to the bliss of it, your face drunk with pleasure—you could forget the whole world in one blessed eruption.†
Chpt 4.10
And quoting the biblical verse "Vengeance is mine," he pointed out that the state, if its purpose was ennoblement and not coercion, should not repay evil with evil, and went on to repudiate the concept of "punishment," after first having refuted that of "guilt," basing his argument on scientific determinism.†
Chpt 6.6 *

vengeance = the act of taking revenge
He had just begun to climb again when—admittedly, just as expected— it began to snow and blow with a vengeance.†
Chpt 6.7

with a vengeance = with intensity

Definitions:

(1) (vengeance as in: vengeance is mine)
the act of taking revenge

(Revenge means to harm someone to get them back for something harmful that they have done.)
(2) (with a vengeance as in: with a vengeance)
with intensity
(3) (meaning too rare to warrant focus)