Geometry of triangles, circles and squares; proportions
Equilateral triangles on squares, circular sectors and inscribed figures, with rules on quantity and number
A densely worked sheet of geometry and proportion, arranged in columns of text and diagrams across two joined pages. Leonardo argues that equilateral triangles built on the sides of squares keep the same proportion as the squares, examines the six equal triangles that fill a circle, and states rules by which equal quantities diminish in size as they increase in number. Diagrams show inscribed hexagons, circular sectors labelled b c a, and a double square and double circle touching one another. In the lower half a large drawing shows a spiral stair climbing over a vaulted arch beneath a rocky crag, with an architectural plan in the corner.
On this page
Triangles on squares keep the squares' proportion
Of equilateral triangles built upon the sides of squares, the proportion from triangle to triangle equals that from square to square. In general, the ratio between similar parts equals the ratio between their similar wholes.
Equal quantities diminish as their number grows
Equal quantities that increase or decrease equally in number grow in magnitude as much as they shrink in number. Thus if a quantity of one hundred is divided into ten equal parts, each part is diminished by 9/10: everything divided into parts increases in number.
Six equal triangles that fill a circle
The six equal triangles that touch one another and also touch the centre and the circumference of one same circle have, without doubt, their three sides of equal length. This is the hexagonal division of the circle into equilateral triangles.
Double square and double circle
Two squares, each double the other, touch one same circle in four places; and two circles, each double the other, touch one same square in four places, giving a nested figure of squares and circles in a 2:1 relation.
Inscribed hexagon and circular sectors
An inscribed hexagon labelled b c a – m n – o – q p accompanies a note on finding the due length of the portion, and two circular sectors labelled b c a – d and f g e – h illustrate that the proportion between portions of circles follows that between the sides of the equilateral triangles in which they sit.
Spiral stair over a vaulted arch
The lower half is dominated by a drawing of a broad spiral or ramping stair winding up over a vaulted arch, surmounted by a rocky mass, with a small architectural plan sketched in the lower-left corner. No text describes it, so it is read from the image alone.