Doubling of Concentric Circles and Squares
Geometric proportions: the right angle as judge of the areas of circles and squares
This geometry page sets out a family of proportion rules for figures sharing a common centre. Two concentric circles are in the ratio of two to one when a square inscribed between them touches both; likewise two concentric squares are doubled when a circle set between them touches both. Leonardo proves the doubling from the eight triangles composing the larger square, and closes with the 'virtue of the right angle' a b c, which acts as judge of proportions among nested portions of circles. The margin carries the corresponding diagrams: circle-in-square-in-circle, a square with diagonals, and a stack of circular segments in an inverted triangle.
On this page
Two concentric circles doubled by an inscribed square
Circles drawn about one and the same centre stand in the ratio of two to one when the square interposed between them is in contact with each of them. The upper diagram shows a circle inside a square inside a circle.
Two concentric squares doubled by an inscribed circle
Conversely, squares made about a single centre are double one another when the circle set between them touches both squares. The middle diagram shows a square with a rotated square and diagonals.
Circle-to-circle as square-to-square, by their diameters
The doubling is proved because, of the eight triangles composing the larger square, the smaller square contains four. The ratio from circle to circle is the same as from square to square, formed by the multiplication of their diameters.
The right angle as judge of proportions (a, b, c)
Of all the portions of circles in contact within the right angle, the greater is always worth all the lesser together; and of the parallels that receive those portions, the greatest contains and is worth all the smaller ones made within the right angle a b c.