Quadrature of lunes and circles: equal-area figures
Semicircles, lunes, quadrants and inscribed squares compared for equal area, with the squaring of a triangle's angle.
The sheet is densely covered with geometric figures — semicircles, quadrants, crescent lunes, and squares inscribed in and circumscribed about circles — each lettered and paired as equal in area (a equals b, c equals d, and so on). The notes assert that a circle touching a square's sides and one touching its angles are double one another, and that comparable squares about a circle are likewise double; Leonardo also proposes to 'square' the angle of the triangle g in the figure n r t. The reasoning turns on halves and eighths of circles that are quadruple one another.
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Equal areas: semicircle and quadrant (a equals b)
At the top, drawn from right to left, a semicircle and a quadrant of a circle are labelled a and b and declared equal in value. Beneath them the same figures with a lune and a portion are labelled c and d and likewise called equal.
Squaring the angle of triangle g
In a cluster lettered g, o, m, p, h, t, r and n, two portions are removed from h and two similar equal ones from m, leaving equal remainders. From g the value of the two portions below and above is taken, and Leonardo then proposes to square the angle of the triangle g, shown below in n r t.
Circles inscribed and circumscribed to a square are double
Along the right margin, figures of inscribed and circumscribed squares support the claim that the two circles of which one touches the sides and the other the angles of the same square are double one to another. This holds because one circle's diameter is the square's diagonal and the other's is the square's side.
Squares touching one circle by sides or angles are double
The converse is stated for squares: the two squares of which one touches a circle with its angles and the other with its sides are double one to another. A note directs the reader to continue toward the left.
Three equal surfaces a, b, c
A circular ring, a circle and a quadrant of a circle are lettered a, b and c and declared equal surfaces among themselves. A neighbouring circle inscribed in a square carries the note that a equals b, c and d together.
Halves and eighths of quadruple circles
One figure states that a equals b because it is the half and the eighth of circles that are quadruple one to another. Another labels its parts 'doubles', asserting that the two parts are worth the remainder.