Origins
The word 'breath' conceals a semantic transformation that reveals something profound about how the early English perceived the act of breathing. It descends from Old English 'brǣþ,' from Proto-Germanic *brēþą, but its original meaning was not 'air in the lungs' or 'the act of respiration.' It meant 'smell,' 'odor,' 'scent,' 'vapor.' Breath was named not for the inhaling but for the exhaling — and specifically for what you could smell on the exhaled air.
This makes historical sense. In a world without germ theory or gas chemistry, the most salient quality of a person's breath was its odor — the smell of what they had eaten, drunk, or the diseases they carried. Old English 'brǣþ' could mean the scent of flowers, the stench of decay, or the vapor rising from a hot spring. The specialized respiratory meaning developed gradually during the Middle English period as the word narrowed from 'exhalation you can smell' to 'exhalation' to 'respiration' to 'the air you take in and push out.'
The further etymology of Proto-Germanic *brēþą is debated. One proposal connects it to a PIE root meaning 'to burn' or 'to warm,' which would make 'breath' cognate with words for heat and vapor — breath as the warm mist that rises from the mouth on a cold day. Another links it to Germanic words for 'haste' or 'quickness' (Old Norse 'braðr' means 'quick, hasty'), though the semantic connection is less clear.
Old English Period
The verb 'breathe' is a Middle English formation derived from the noun 'breath' by adding the verbal suffix — it is not the other way around. Old English used 'ēþian' (to breathe, from 'ēþ,' easy, comfortable — breathing as 'being at ease') or 'orþian' for the act of respiration, not a derivative of 'brǣþ.' The verb 'breathe' gradually replaced these older terms.
Across world languages and philosophies, 'breath' words frequently double as 'spirit' or 'soul' words. Latin 'spīritus' (spirit) comes from 'spīrāre' (to breathe). Greek 'pneuma' (πνεῦμα) means both 'breath' and 'spirit.' Sanskrit 'ātman' (soul, self) is related to German 'atmen' (to breathe). Hebrew 'ruach' means both 'wind' and 'spirit.' The equation breath = spirit = life is one of the most universal conceptual metaphors in human language. English 'breath' does not directly participate in this equation (English uses the Latin-derived 'spirit' and 'respiration' for the elevated senses), but the underlying intuition — that breath is the visible sign of life — pervades English idiom: 'breath of life,' 'breathless with excitement,' 'to draw one's last breath,' 'don't hold your breath.'
The compound 'breathtaking' (so impressive as to leave one gasping) dates from the mid-nineteenth century. 'Breathless' has been in use since Old English. 'Short of breath' as a medical descriptor appears in the sixteenth century.