Derived Shadows and the Squaring of Curved Figures
Light-and-shadow studies joined to the quadrature of circular portions
Two mounted fragments carry two intertwined investigations. The upper strip is dense with ray diagrams of luminous spheres and shadow-casting bodies, establishing that a derived shadow is darker at its extremes than in its middle, that its darkness depends on how much darkness each part sees, and that an opaque surface takes the color of its object. The lower sheet turns to geometry: starting from a hexagon and its circular portions, Leonardo works to square a curvilinear figure by removing all curved sides, transporting portions a b c into squares q f s t and e p d n, and comparing two concentric circles of quadruple proportion. He concludes that six great portions of the larger circle equal twenty-four small portions of the smaller, all built on the semidiameter or the sixth part of the circumference.
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Where the greatest brightness lies
The nearer the opaque body is to the light, the more remote the greatest brightness stands from the greatest derived shadow around it. The boundary of that greatest derived shadow is darker than its middle.
Darkness proportioned to the darkness seen
The part of a derived shadow that sees the greater sum of darkness is the darker, and the part seeing less darkness is lighter. The surface of every opaque body partakes of the color of its object, and a uniformly transparent medium lets every color and figure pass without occupying place within it.
Darker at the extremes than the middle
a b sees the end of shadow c d and is faint; c sees the whole shadow c b and is deepest; n m sees half and is of middling darkness — proving the greatest derived shadow darker at its edges than at its center. The pyramid S d e is tinged with the colors of its objects and varies in darkness as it sees more or less of the dark object c b.
Squaring a curvilinear surface
To square a curved figure, all its curved sides must be removed: stripping the six portions from the first figure leaves a hexagon, which is squarable. Portion a b c is carried into a square q f s t, divided into four parallels; one parallel becomes square e p d n by the last proposition of Euclid's second book, whose straight side serves as the semidiameter of a circle receiving twenty-four equal portions.
Two concentric circles in quadruple proportion
Two circles, one within the other, stand in quadruple proportion, so the great portion a b c is worth four of the small portions like d e f. The six great portions of the outer circle therefore equal the twenty-four small portions of the inner one, since circle is to circle as square is to square made by the diameter multiplied by itself.
Proportion preserved by adding or removing parts
If from things of a given proportion one removes a part of like proportion, the remainder keeps the same proportion; and if one adds things of the same proportion, the quantities keep it too. The paired figures 2, 4 and 1, 2 stand beside the rule.