Geometry of lunules, octagons, and the quadrature of figures
Studies in squaring curved figures, with a cone and cylinder built on one circle
On blue paper Leonardo covers a sheet with quadrature studies: crescent-shaped lunules, octagons, divided double circles, semicircles and a cone-and-cylinder pair, all lettered for demonstration. The notes seek to transform curved surfaces into equal rectilinear ones, stating for instance that a lunar crescent is equal to a parallelogram and that a cylinder's surface can be made equal to a circle double a given cone's surface. Several blocks are worked through arithmetically, taking equal parts from two quantities and adding to a third. An inserted rectangle at lower left carries a further block of upright script.
On this page
The lunar crescent equal to a parallelogram
Within a quadrilateral bounded by two curved sides, Leonardo asserts a quadrature result: the lunar crescent (falcata lunare) is equal to the parallel-sided figure. This is one of a family of lunule figures drawn across the sheet, each pairing a curved sliver against a rectilinear equivalent.
A cone and cylinder on one and the same circle
If on a single circumference a cone and a cylinder are raised so that the cone's slant equals the cylinder's height, then the cylinder's surface is made equal to a circle double the surface of that cone. The claim relates the lateral surfaces of the two solids through a common base circle.
Three equal quantities redistributed
Set beside three octagons labelled a, b, c: if there are three equal quantities and one takes equal parts from two of them and adds them to the third, then removes as much again from that third, the remainder equals one of the other surfaces plus a part. The reasoning is called out as done 'by algebra'.
Squaring two surfaces and taking their difference
Four figures along the lower margin, lettered a, b, d, c, e, f, g, h. Having surface a b equal to surface c d e, Leonardo squares a b and sets it at f, squares b c d e and sets it at g, then takes g from f, and h remains. The construction reduces the comparison of two areas to a single residual figure.