What is meant by "loose the loins of kings" in Isaiah 45:1?
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The term "loins" (in Hebrew, מתן) throughout the Bible often refers to some sort of strength when the loins were girded (see 2 Kings 4:29, for example), so loosening their loins would mean to weaken them. This could be interpreted completely in the figurative sense (simply making them weak, or removing them from power, how most translations render it), or it may be taken a bit more literally, and be understood as disarming these kings, or removing their weapons, as many of the translations here understand (see NIV, CSB, etc.).
This isn't exactly polite, and I do apologize if it is deemed offensive, but Bible teacher Chuck Missler addresses this phrase this way:
In Isaiah 44 and 45, God not only describes the ease with which Cyrus would enter the city with the "two leaved gates" (gates that were not even shut against the invaders!) but also notes that He would "loose the loins of kings" before Cyrus – a euphemism regarding the fear these kings would feel and the mess in their pants they'd make as a result.
-- Isaiah and Cyrus the Great (Koinonia House)
It seems, then, that the loosing or weakening of loins may refer to a physiological phenomenon which accompanies extreme (and in Belshazzar's case, warranted) fear.