Geometry: Squaring the Circle with Irregular Lunes
Circles divided into irregular lunes and two-cornered spirals, shown equal to a square
Headed 'Geometry', this folio pursues the squaring of the circle through irregular lunes (lunole) and two-cornered spirals (eliche bisangole) drawn down the right margin. Leonardo argues that a two-cornered spiral surface a d b c o, turned by motion, becomes a rectangular surface equal to a quarter-circle, and that a circle can be divided into equal lunes 'by the rule with which the roots of each number are made'. He observes that such irregular lunes are infinite in their variety of curvature. The dense mirror-script at left carries his running demonstration.
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Irregular lunes dividing a circle
A circle is broken into crescent shapes labelled with the points d, b f, e c and o, a c, d b, one of which Leonardo names 'an irregular lune' (lunola inregolare). He treats these curved-sided divisions as building blocks for equating curved and rectilinear areas. He adds that lunes of irregular sides are infinite in number, of infinite variety of curvature.
Two-cornered spiral equal to a quarter-circle
Under the heading 'Principal column', a two-cornered surface of irregular sides and unequal curvature, its concavities turned toward one centre, is set equal to a square. The sought spiral a d b c o becomes, by motion, a rectangular surface equal to a quarter of a circle; the figure above is first divided into 4 like spirals, one being b e c n d, and into 4 triangles, one being b d f, whose curved side by its motion yields a straight line equal to that curved side.
Dividing a circle into equal lunes
Given a point on a circle's circumference, all the equal parts into which the circle can be divided are joined to it, producing circles that each exceed the next by an equal excess. Leonardo says these are made 'by the rule with which the roots of each number are made', and asks for a required number of equal lunes together worth a whole circle.