Turning a Cube into a Cylinder of Given Thickness
Finding the cylinder's length by set-square, hypotenuse and compass arcs
Continuing the series on transforming solids, this proposition takes a cube a and a cylinder of a given thickness b c and seeks the cylinder's length. Where the previous page fixed a length and left the thickness uncertain, here the thickness is known and the length is constructed geometrically: a set-square (squadra) with side d o is slid along the line d o, a hypotenuse d r is extended, and two compass arcs m s t and d v q are drawn until the circle becomes tangent to the set-square. The length is determined where the arc, its chord and the corner of the set-square meet at q. A semicircular arc with a right-angle set-square is sketched at the top of the leaf.
On this page
Constructing the cylinder's length from its thickness
With the cube a and the cylinder's given thickness b c, the cylinder is produced to an indefinite length b c f e together with the hypotenuse d r. The set-square d o g is set with side d o on line d e and the compass foot placed at m p; the arc m s t is drawn and the set-square shifted until the circle is tangent to side o q, then a second arc d v q fixes the length at q.
Correcting the trial position
The construction is iterative: if the circle should touch the hypotenuse before the set-square, Leonardo increases the trial space, sliding the ruler back and forth until arc, chord and set-square coincide, so that the sought length is read exactly at the set-square's corner.