Geometry of the semicircle, cylinder and cone
Circle proportions and cone-cylinder surfaces, with tables of ratios
On these facing pages Leonardo works through relations among circles, cylinders and cones. He argues that the periphery of a semicircle equals the circumference of a circle worth half the semicircle, that the largest triangle drawn from a square is half of it, and that a cylinder's surface is double that of the largest cone of equal length standing on the same circular base. Interleaved on folio 198r are tables of ratios and repeated subtractions of four, reducing fractions such as 12/14 to 6/7.
On this page
Periphery of a semicircle equals a circle's circumference
Leonardo claims that the curved periphery of a semicircle equals the circumference of a circle worth half of that semicircle. He labels the semicircle s a o with periphery o n a h s and, carrying its ends o s to p, generates a circle a n p r worth half the semicircle and a quarter of the whole circle a o v s. He grounds it on the rule that circles are to one another as the squares of their diameters.
The largest cylinder drawn from a cube
A marginal note states that the largest cylinder that can be cut from a cube has its rectangular (lateral) surface equal to four circles like its circular face.
Cone and cylinder: surface relations
When a cone and a cylinder stand on the same circle with sides of equal length, the cylinder's surface is double that of the cone, quadruple the cylinder's base and double the cone's lateral surface. Leonardo generalises that any cylinder has double the surface of its largest cone of equal length. He also gives the largest triangle from a square as half the square.
Ratios reduced to lowest terms
A row of paired ratios reduces fractions to lowest terms: 12/14 to 6/7, 30/34 to 15/17, 32/36 to 16/18 and 8/9, 36/40 to 18/20 and 9/10.
Intrinsic space as portions of the circle
An inverted note explains that the intrinsic space consists of two portions worth four extrinsic proportions of the whole circle, tied to the surrounding circle diagrams.
Subtracting four: a table of remainders
Under the heading that a b equals c d, Leonardo tabulates the result of taking 4 from a run of numbers: 4 from 4 leaves 0, from 6 leaves 2, from 8 leaves 4, and so on up to 4 from 20 leaving 16/20 or 4/5.