Some English idioms don’t necessarily make all that sense. What on Earth does it really mean to be raining cats and dogs, after all. At the same time, there are other idioms that are reasonably logical. If you break them down, striking a chord or proving your mettle sound about right. And the same can be said when you’re talking about a hair’s breadth.
And it makes a lot of sense that we talk about a hair’s breadth and not a hair’s breath. For one, they’re not pronounced the same way, but you can see (or hear) how the difference can be overlooked (or overheard) in casual speech, resulting in incorrect spelling when the term is written. “Breadth” generally refers to the measurement of something from one side to the other, which can also be called the width.
While you may have hair of varying length, it is generally accepted that the width of a single hair is going to be pretty small. And so, when you say that the margin of difference is just a hair’s breadth, you are saying that the margin or distance is almost imperceptibly small. Given this, it makes no sense to refer to a hair’s breath, because your hair doesn’t exactly breathe.
Then again, if you mistype “hair” with “hare” (the rabbit-like animal), then you could logically refer to a hare’s breath or a hare’s breadth, because the long-eared creature does indeed exhale and it does indeed have a certain width to it. However, neither of those is the idiomatic term commonly used in English. Just stick with a hair’s breadth.
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