"....the length and breadth of the room being known, we will find a number that has the same proportion to the breadth as the length has to the number sought.......if the place we intend to vault is nine feet long and four feet wide, the height will be six feet"
In a Geometrical Mean the first amount is in proportion to the second amount as the second is to the third. a is to b as b is to c. Or a:b = b:c. In Palladio's example;
6 exeeds 4 by a third of 6 which is 2,
just as 9 exeeds 6 by a third of 9 which is 3.
Or 4:6:9. Or 4:6 = 6:9.
Practically this means, in the words of Palladio;
"..we find this by multiplying the lesser extreme with the greater; because the square root of the number which will result from such a multiplication will be the number we seek."
In his example we multiply the lesser extreme, or width, which is 4, by the greater extreme, which is 9, to get 36. The square root of 36, (i.e. the only number which when multiplyed by itself will give 36) is 6. Thus the height of the room is 6.