Shadows cast by a sphere, and the squaring of the lune
Shadow projection studies above; below, dividing one circle into parts proportional to another
Turned on its side at the upper right, a run of small diagrams works out how a spherical body throws 'mixed' and 'simple' shadows onto a wall, arguing that only a point-like light can cast the sphere's true shadow and that the derived shadow lengthens or shortens with distance from the light. The lower and left portions are crowded with geometrical figures — superimposed circular sectors, triangles and a semicircle over a quadrant — through which Leonardo attacks the squaring of the lune (falcata) and the semicircle. Long verbal rules then explain how to take a part of a larger circle equal to a given part of a smaller one, supported by rule-of-three arithmetic and columns of fractions. Both halves are packed with untranscribed numeral jottings that accompany the figures.
On this page
A sphere's true shadow requires a point-like light
The diagrams distinguish a 'mixed shadow' flanking a central 'simple shadow'. No elongated light source, the note argues, can cast onto the walls the true form of the separate shadows of spherical bodies; only when the centre of the light is equidistant from the extremities of the body can the shadow imprint the body's real shape.
Derived shadow shortens with distance from the light
Two parallel statements set out the same law: shadow-casting bodies make their derived shadow more or less short according to whether they stand nearer to or farther from their light. The accompanying figures track cones of shadow springing from small spheres.
Squaring the lune and the semicircle
A semicircle set on a quadrant of a circle demonstrates a 'method of squaring the lune (falcata), and consequently the semicircle': the three pieces of the semicircle are reduced to triangles together with the three pieces of the lune, completing the three triangular pieces it lacks. The construction is labelled b a.
Finding equal parts by subtracting sectors
In the central column two superimposed sectors are compared: a and b joined together are equal to c, so c exceeds b by the whole quantity of a; subtracting b from c leaves a remainder equal to a. A companion figure works a 'doubled pyramid' so that removing c from b leaves a square part equal to a.
Taking a proportional part of the larger circle
The left column gives a verbal rule: to take a part of the larger circle equal to a given part of the smaller, multiply the smaller circle's part by the denomination arising from the proportion between the circles. Worked with a circle quadruple a smaller one, this yields, for example, 4 times 4 makes 16.
Fraction table of quadruple proportions
A column pairs unit fractions of the smaller circle with their quadruple equivalents: 1/2 with an eighth, 1/3 with 1/12, 1/4 with 1/16, 1/5 with 1/20, 1/6 with 1/24. The rule concludes that, in a circle quadruple a smaller, 1/12 of the larger equals 1/3 of the smaller, since 3 times 4 makes 12.