Removing weighted pieces of a pyramid by squared fractions
The right triangle q p o m n a subdivided 1, 3, 5, 7, with a rule linking the cut to the square of the root.
The page returns to the divided right triangle, its apex labelled q and its sides marked p, o, m, n and a, subdivided into 1, 3, 5 and 7 similar triangles whose "excesses" are said to be equal. Leonardo gives a rule for cutting off a chosen weight: multiply the root of the desired portion by itself and remove that from a corner, so a quarter of the height gives one-sixteenth of the pyramid, a half gives one-quarter, and three-quarters gives nine-sixteenths. He closes with the curved pyramid, whose ends meet in the finishing of its circles, warning that even over thousands of miles it could never close except by the method he has given.
On this page
Square the root of the portion you want
To take a wanted amount of the pyramid, multiply by itself the root of that number and remove it from any corner. The worked examples give one-sixteenth, one-quarter and nine-sixteenths of the whole.
Cutting a quarter, a half, and three-quarters
Removing a quarter of the base, matching a quarter of the length, gives four times four, so one-sixteenth of the pyramid; half of the base with half the length gives one-quarter; three-quarters of the base with three-quarters of the length gives nine-sixteenths.
The curved pyramid closes in its circles
The curved pyramid finds its end in the finishing of its circles. Even if it ran thousands of miles to join with itself, those circles could never be completed except by the method given.