Areas of circles as squares of diameters, and lantern studies
Geometry of nested circles with octagonal cupola sketches, pinwheel ornaments and sums
In lettered figures and a marginal note Leonardo proves that the ratio of one circle to another equals the ratio of the squares of their diameters, so that a circle of double diameter is quadruple in area and a great circle divides into two circular and two curvilinear-triangular parts of equal value. He observes that two equal touching circles occupy the least space within a circumscribing circle. The sheet also carries prominent perspective sketches of octagonal domed lanterns or cupolas, pinwheel and interlace ornaments set in circles and on a footed goblet, and columns of figures in soldi and denari. The architectural and ornamental drawings are not covered by the transcription.
On this page
Two touching circles within a circumscribing circle
A marginal proposition states that two circles touching one another and touched by the circumference of a surrounding circle occupy more space within it the more they differ in size, and least space when most similar. This is the extremal case underlying the lettered figures below.
Circle areas as squares of diameters; great circle quadruple the small
Multiplying the diameter of circle a b by itself yields the square a b c d, and doubling that diameter into b e yields the square a b S t, four times larger; so the greatest circle equals four times the smallest circle a b. Removing the two lesser circles leaves the greatest circle divided into four equal parts, two circular and two curvilinear-triangular.
Octagonal domed lanterns in perspective
Two carefully shaded octagonal domed pavilions or cupola lanterns, columned and pinnacled, are drawn in perspective at the centre of the sheet. They are not part of the transcription and belong to Leonardo's architectural studies of lanterns and cupolas.
Pinwheel and interlace ornaments in circles
Several circles are filled with pinwheel and triquetra-like interlace, a quatrefoil is set in a square, and a footed goblet bears a spinning pinwheel motif. These decorative figures appear among the geometry but are not transcribed.
Columns of figures in soldi and denari
Below and beside the circle figures Leonardo sets down columns of numbers together with sums in soldi (shillings) and denari (pence). These calculations sit alongside the proportional proofs of circle and square areas.