I'm familiar with blessings/benediction given after sneezing in European and Near Eastern cultures, and they are old old old. One of the Arabian Nights or 1001 Nights tales involves a teacher who makes his pupils cross their arms and wish him good health after every sneeze. (The sneezing and good health wishes are padding on the core story, which is a tale type known as Bringing Water from the Well and in this case turns on the command for the students to cross their arms). This connection is significant because both medieval=>modern cultures in question rooted their original medical worldview in the writings of the ancient Greeks.

It's there that we find what sets sneezing apart. Namely, according to Galen, sneezing is directly connected to the head, to the brain. It is restorative in most cases: it allows excess steam and "mucus", a substance which Galen is careful to distinguish from the phlegm of chest-coughing, to escape. However, Galen goes on, sneezing while someone has a chest-cold rather than some other form of disease or no disease at all (as from a dusty room) is actually harmful, because it irritates the chest as well, making it prone to worsening illness. This passage is picked up by al-Maliki and actually makes it into some surviving "medical textbooks" from the medieval Islamic world.

So there are two features in play here: one, the connection to the brain; two, the idea that sneezing can be either good or bad. Both of these ideas bleed out from medical writing into "popular culture", or more likely, both of them bled into medical writing from surrounding ideas. (It's not like Galen made up medicine).

Sneezing as portent is all over ancient writing from the Mediterranean world. In the Odyssey, for goodness' sake, Telemachus sneezes (book 17) and Penelope rejoices in it as a sign that all her suitors will be killed. One of the love spells in the Egyptian-Greek magical papyri has the lovesick man recite a long passage about Isis, Thoth, and Osiris. And what happens while the man is reading matters:

If a large amount of saliva forms in your mouth as you speak understand that she is distressed/lovesick and wants to talk with you; if you yawn frequently she wants to come to you. But if you sneeze two times or more, she is in good health and is returning to where she lives; if you have a headache and are crying, she is distressed or even dying.

The connection with the brain becomes a connection with a person's life force--their soul, if you will. In Jewish folklore, sneezing is the first sign of a person temporarily brought back to life; in the medieval Christian west, sneezing aids mothers in childbirth. What's especially noteworthy in the second example is that the connection between sneezing and good health is usually explicitly pointed out by someone in the literary anecdote as a blessing. Although I've not seen a direct reference to the common modern belief that "bless you" is a way to stuff someone's soul back in them after they expel it in a sneeze, there are parallels in early modern literature: the idea that souls can be exchanged by mouth, as in a kiss.

However, just like sneezing can be a bad sign medically (according to Greek medicine that the medieval/early modern world inherit), sneezing is not always a good omen. Christian Friedrich Garmann's ~1670 work On the Wonders of the Dead, which collects contemporary quasi-scientific and superstitious ideas about corpses and how they got that way, observes that multiple days of sneezing in a row, the sneezes occurring in the same place, are said to forecast death for someone in the house.

So there's not a lot of "when" to the question, but there is a whole range of "why". Mediterranean cultures back and back and back have connected sneezing with the possibility of good and bad health, good and bad fortune, and the desire to affect its course.

Furthering the question: why not with coughing, which is so similar?

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