"...let the room to be vaulted be twelve feet long and six broad; add six to twelve and it will make eighteen, the half of which is nine; the vault ought therefor to be nine feet."
In an Arithmetic Mean, the second amount exeeds the first by the same amount as the third exeeds the second, as in 2:3:4. Three exeeds two by the same amount that four exeeds three. Or, in Palladio's example:
9 exeeds 6 by 3,
which is the same amount by which 12 exeeds 9.
Practically, this means taking the length and adding it to the width, then dividing the result in half, as Palladio described.