Squaring Lunes and Sectors; Proportion of Triangles
Transforming curved figures into equal rectilinear ones, and comparing triangles by their bases
This densely worked sheet is crowded with geometric figures—semicircles, circle quadrants, sectors, boat-shaped lunes, triangles and annular rings—with notes on transforming and 'squaring' curved areas into equal rectilinear ones. Leonardo repeatedly seeks to square a lune or sector against a triangle, tracking the parts by letter labels (a, b, c, d, e, m, n). He also states the two proportion rules he is using: triangles of equal height stand in the ratio of their bases, while similar triangles stand as the square of their bases. Beyond the transcribed blocks the leaf carries further faint figures and writing that are not included here.
On this page
Squaring the lune c d e
Leonardo sets a b equal to c d, then aims to square the lune c d e by removing the squaring of n m, which equals the squaring of c d. The remainder of n m would then equal the sector figure a b. He notes that a quarter and an eighth of circles are quadruple to one another.
Triangles in the ratio of their bases
The proportion of triangle b to triangle a is the same as the proportion of the bases of those triangles. This holds because the triangles have equal heights, being drawn between parallel (equidistant) lines.
Similar triangles as the square of the base
For similar triangles the proportion from triangle to triangle is as from base to base multiplied by itself. This marginal note fixes the rule that similar figures scale as the square of their linear measure.
A triangle worth four smaller triangles
Triangles a d c and c a e, having equal base and height, are equal, and each is worth the two triangles d a e and d c e. The triangle a n c is worth 4 triangles d n e, being twice as tall and twice as wide, and is worth twice the triangle c n e.
Reshaping a ring without changing its area
A curvilinear 'parallel' a has its arcs equal and similar. Leonardo wants, from the same quantity, to make a parallel whose arcs belong to circles double one another, without diminishing the original area.