Geometry of lunes: proportional division of circles
Squaring curved portions by the quaternary number, with a note that mountains are made and unmade by water
A densely written sheet crowded with geometric figures — circles, triangles, squares inscribed in circular rings, lunes, and a semicircle packed with small tangent circles. Leonardo works out the proportional division of circular portions and lunes, repeatedly showing that a smaller circle is a quarter (or a sixteenth) of a larger one and that curved figures can be divided to infinity by the 'quaternary number.' Marginal jottings add an 'Elements' heading and the geological aphorism that mountains are made and unmade by the course of the waters. The central and right columns carry further continuous prose beyond the labelled figure captions transcribed here.
On this page
Circles, triangles and portions in the same proportion
Since the smallest circle is a quarter of the largest, the portion removed from the smallest is a quarter of the portion removed from the largest. The largest triangle resolves into 4 smallest, and the largest circle is worth 4 smallest circles, so triangle and circle keep the same ratio. The greatest portion resolves into 4 equal parts a b c d.
Sub-quadruple and sub-sixteenfold circles
The circle built on the semi-diameter of another circle is one quarter of it. If instead the smaller circle takes the diameter of a quarter of the larger circle's diameter, it becomes one sixteenth of the greater circle. The three internally tangent circles illustrate the nested ratios.
Division to infinity by the quaternary number
The semicircle can be divided to infinity into equal parts, because two smallest equal circles set within one circle always contain half of it, the division being always one quarter. The greatest portion is likewise stated to be divisible to infinity by the quaternary number.
Turning a curved figure into a square
First remove the two portions e and f, filling a b in imagination since a b is worth as much as e f. Then remove the upper convexity and use it to fill the lower concavity, and there remains a square like the figure M. A worked recipe for reducing a curvilinear figure to an equivalent square.
Mountains made and unmade by water
A marginal aphorism at the head of the sheet: the mountains are made and unmade by the course of the waters. It stands apart from the surrounding geometry as one of Leonardo's recurring reflections on erosion and the shaping of the earth.