Squares Inscribed in Circles: Areas and Proportions
Red-chalk geometry with a spiral volute and a small window frame; columns of figures.
A red-chalk (sanguigna) sheet of geometry exploring squares inscribed in and circumscribed about circles together with their proportional areas: the square set inside the larger circle is 9/16 of that circle, and the largest circle inside a square is 4/5 of the square. A pen note assigns capacities of 100 and 50 braccia to two circles and 64 to a square, concluding that the corners a b c d amount to 14 braccia. Around the figures run columns of numbers, some written with the sheet turned on its side, along with a spiral volute, a small window or aedicule, and several triangles.
On this page
Square inscribed and circumscribed with a circle
The square that can be drawn inside the larger circle is 9/16 of the whole circle, while the largest circle that can be drawn inside a square is 4/5 of that square. From this Leonardo concludes that a b c d is the fifth part of the circle f. The figure is lettered m, b, a, o, f, n, c, b, r.
Capacities of the circles and square
In pen, the larger circle is given a capacity of 100 braccia, the smaller 50, and the square 64. It follows that the corners a b c d come to 14 braccia. The note breaks off mid-sentence at 'and the larger'.
Columns of figures
Several stacks of numbers are set beside the diagrams, one group headed simply 'Numbers' and another written with the sheet turned onto its left side. They appear to work through the areas and ratios of the figures above.
Spiral volute and window frame
Among the geometric figures Leonardo sketches a large spiral volute, resembling the scroll of an Ionic capital, and a small framed opening like a window or aedicule with a pediment. These architectural motifs share the sheet with the mathematical studies.