Quadrature studies: squaring circles, rings and lunes
Concentric circles, inscribed squares and lunes, with a window elevation and a vault
A sheet crowded with figures for the quadrature of the circle: concentric circles with inscribed squares, circular rings (corone), lens-shaped 'bisangoli' and lunes set within squares, arranged in three columns of demonstrations. The accompanying notes argue by equal areas and Euclidean common notions ('if from equals you take equals, the remainders are equal') that rings, lunes and portions of circles can be converted into rectilinear squares and crosses. At upper left, untranscribed drawings show an architectural window or portal elevation with columns and a groin vault; a small axe or halberd head appears at lower left.
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A circular ring equals the circle it encloses
Concentric circles and inscribed squares are labelled c, d, b, e, K, a, f, h, g. Leonardo argues that a b cd e f g h is worth K squared, because the circular ring equals the circle it surrounds and its eight portions are worth the four of the smaller circle. Removing equals from equals leaves the remainders equal.
Euclidean common notions on equal and similar figures
A marginal note states that taking equal and similar figures from equal and similar ones leaves an equal and similar remainder, while taking equal and dissimilar ones leaves remainders that are equal but not similar. Elsewhere he adds that if from a known quantity a known quantity is removed, the remainder is known. These axioms underwrite the area-swapping proofs across the sheet.
Filling four lunes to turn a curved cross into a straight one
Working with lunes inscribed in a square, Leonardo converts the four greatest portions into eight half-portions. Packing those eight into the four concavities of the four lunes turns the curvilinear cross into a rectilinear cruciform surface.
Bisangoli (lens-shapes) and the squaring of the circle
Two circles set within an inscribed circle are labelled f, e, K, a, L, n, p, d, b, i, g. Leonardo declares the four lunes squarable, notes the minor circles are each half the greatest circle, and reasons that the smallest circle L a K equals triangle d p, so that the lens-shape a is worth p d once equal parts are removed from each.
Window elevation and groin vault (untranscribed drawings)
The upper-left corner carries pen sketches unrelated to the geometry: an arched window or portal in elevation flanked by columns, with a small groin (cross) vault drawn just above it. These architectural studies bear no accompanying transcribed text.