Reducing a Cylinder's Length: How It Thickens
Set-square-and-arc method for the equivalence of cylinder and cube
This page continues Leonardo's geometry of the equivalence of solids, showing how a set-square (squadra), a hypotenuse and a compass-drawn arc are worked back and forth to fix a sought length. The main note treats how a long cylinder a b c d, when shortened to a given length c n, must grow thicker, and prescribes first reducing the cylinder to its equivalent cube and then redistributing that cube into the new length to read off the thickness. A companion note treats the inverse case: a short cylinder stretched to a given length, and how much it narrows. A large semicircular arc drawn against a right-angle set-square occupies the upper right of the leaf.
On this page
Fixing the true length with set-square and arc
Continuing a construction, Leonardo adjusts the space m p to enlarge or shrink a described arc. If the circle touches the set-square before the hypotenuse he diminishes m p for a smaller arc, and working in this way he fixes the true length sought.
Making a long cylinder shorter
Let cylinder a b c d be the longer one and c n the given shorter length. To learn how much thicker the shortened cylinder becomes, reduce the long cylinder to its equivalent cube by the first rule, then by the second redistribute that cube into the given length to find the required thickness.
Stretching a short cylinder to a given length
The inverse problem: a short cylinder extended to a given length, and how much it narrows. Reduce the first cylinder to its cube by the first proposition, and by the second extend it to the given distance.